QuestionsAnswered.net
What's Your Question?
Resources to Help You Solve Math Equations
Whether you love math or suffer through every single problem, there are plenty of resources to help you solve math equations. Skip the tutor and log on to load these awesome websites for a fantastic free equation solver or simply to find answers for solving equations on the Internet.
Stand By for Automatic Math Solutions at Quick Math
The Quick Math website offers easy answers for solving equations along with a simple format that makes math a breeze. Load the website to browse tutorials, set up a polynomial equation solver, or to factor or expand fractions. From algebra to calculus and graphs, Quick Math provides not just the answers to your tough math problems but a step-by-step problem-solving calculator. Use the input bar to enter your equation, and click on the “simplify” button to explore the problem and its solution. Choose some sample problems to practice your math skills, or move to another tab for a variety of math input options. Quick Math makes it easy to learn how to solve this equation even when you’re completely confused.
Modern Math Answers Come From Mathway
Mathway offers a free equation solver that sifts through your toughest math problems — and makes math easy. Simply enter your math problem into the Mathway calculator, and choose what you’d like the math management program to do with the problem. Pick from math solutions that include graphing, simplifying, finding a slope or solving for a y-intercept by scrolling through the Mathway drop-down menu. Use the answers for solving equations to explore different types of solutions, or set the calculator to offer the best solution for your particular math puzzle. Mathway offers the option to create an account, to sign in or sign up for additional features or to simply stick with the free equation solver.
Wyzant — Why Not?
Wyzant offers a variety of answers when it comes to “how to solve this equation” questions. Sign up to find a tutor trained to offer online sessions that increase your math understanding, or jump in with the calculator that helps you simplify math equations. A quick-start guide makes it easy to understand exactly how to use the Wyzant math solutions pages, while additional resources provide algebra worksheets, a polynomial equation solver, math-related blogs to promote better math skills and lesson recording. Truly filled with math solutions, Wyzant provides more than just an equation calculator and actually connects you with people who are trained to teach the math you need. Prices for tutoring vary greatly, but access to the website and its worksheets is free.
Take in Some WebMath
Log onto the WebMath website, and browse through the tabs that include Math for Everyone, Trig and Calculus, General Math and even K-8 Math. A simple drop-down box helps you to determine what type of math help you need, and then you easily add your problem to the free equation solver. WebMath provides plenty of options for homeschoolers with lesson plans, virtual labs and family activities.
Khan Academy Offers More Than Answers
A free equation solver is just the beginning when it comes to awesome math resources at Khan Academy. Free to use and filled with videos that offer an online teaching experience, Khan Academy helps you to simplify math equations, shows you how to solve equations and provides full math lessons from Kindergarten to SAT test preparation. Watch the video for each math problem, explore the sample problems, and increase your math skills right at home with Khan Academy’s easy-to-follow video learning experience. Once you’ve completed your math video, move onto practice problems that help to increase your confidence in your math skills.
MORE FROM QUESTIONSANSWERED.NET
Solving Trigonometric Equations
Solving trig equations is finding the solutions of equations like we did with linear, quadratic, and radical equations, but using trig functions instead. We will mainly use the Unit Circle to find the exact solutions if we can, and we’ll start out by finding the solutions from \(\left[ 0,2\pi \right)\). We can also solve these using a Graphing Calculator , as we’ll see below . Note that we will use Trigonometric Identities to solve trig problems in the Trigonometric Identity section .
Important Note: There is a subtle distinction between finding inverse trig functions and solving for trig functions . If we want \(\displaystyle {{\sin }^{{-1}}}\left( {\frac{{\sqrt{2}}}{2}} \right)\) for example, like in the Inverse Trigonometric Functions section , we only pick the answers from Quadrants I and IV , so we get \(\displaystyle \frac{\pi }{4}\) only. But if we are solving \(\displaystyle \sin \left( x \right)=\frac{{\sqrt{2}}}{2}\) we get \(\displaystyle \frac{\pi }{4}\) and \(\displaystyle \frac{{3\pi }}{4}\) in the interval \({\left[ {0,2\pi } \right)}\); there are no domain restrictions . In these cases, we want all solutions in the given interval.
Solving Trigonometric Equations Using the Unit Circle
Let’s start out with solving fairly simple trig equations, and getting the solutions from \(\left[ 0,2\pi \right)\), or \(\left[ {0,{{{360}}^{o}}} \right)\). Here is the Unit Circle again so we can “pick off” the answers from it:
Notice that we always isolate the trig function, and some solutions may have none, or more than one solution. If there are multiply angles on the unit circle for that trig function, and an expression is involved, we may have to divide up the equation into two separate equations and solve each, like the example with \(\displaystyle \theta +\frac{\pi }{{18}}\).
If a square is involved, when we take the square root, we have to include both the positive and negative values. (Note that \({{\left( \cos \theta \right)}^{2}}\) is written as \({{\cos }^{2}}\theta \), and we can put it in the graphing calculator as \(\boldsymbol{\cos {{\left( x \right)}^{2}}}\) or \(\boldsymbol {{{\left( {\cos \left( x \right)} \right)}^{2}}}\)).
Note that sometimes you may have to solve using degrees \(\left[ {0,{{{360}}^{o}}} \right)\) instead of radians. The last problem involves solving a trig inequality .
Solving Trigonometric Equations – General Solutions
Since trig functions go on and on in both directions of the \(x\)-axis, we’ll also have to know how to solve trig equations over the set of real numbers ; this is called finding the general solutions for these equations. We still use the Unit Circle to do this, but we have to think about adding and subtracting multiples of \(2\pi \) for the sin , cos , csc , and sec functions (since \(2\pi \) is the period for them), and \(\pi \) for the tan and cot functions (since \(\pi \) is the period for them). We can do this by adding \(2\pi k\) or \(\pi k\) where \(k\) is any integer (positive, negative, or 0 ); sometimes this can be simplified.
We need to be careful about domain restrictions with our answers. For tan , cot , csc , and sec , we have asymptotes, and if our answer happens to fall on an asymptote, we have to eliminate it.
Here are examples; find the general solution, or all real solutions for the following equations. Note that \(k\) represents all integers \(\left( k\in \mathbb{Z} \right)\). Note also that I’m using “fancy” notation; you may not be required to do this.
Note that you can check these in a graphing calculator (radian mode) by putting the left-hand side of the equation into \({{Y}_{1}}\) and the right-hand side into \({{Y}_{2}}\) and get the intersection. You won’t get the exact answers, but you can still compare to the exact answers you got above.
Solving Trigonometric Equations with Multiple Angles
We have to be careful when solving trig equations with multiple angles, meaning there is a coefficient before the \(x\) or \(\theta \) (variable). This is because we could have fewer or more solutions in the Unit Circle , and thus for all real solutions when we add the \(2\pi k\) or \(\pi k\). Thus, when we solve these types of trig problems, we always want to solve for the General Solution first (even if we’re asked to get the solutions between 0 and \(2\pi k\)) and then go back and see how many solutions are on the Unit Circle (between 0 and \(2\pi k\) ).
When solving trig equations with multiple angles between 0 and \(2\pi \), we’ll typically get fewer solutions if the coefficient of the variable is less than 1 , or more solutions if the coefficient of the variable is greater than 1 . As an example, we typically get two solutions for \(\cos \left( \theta \right)\) between 0 and \(2\pi \), so for \(\cos \left( 3\theta \right)\), we’ll get 2 times 3 , or 6 solutions. As another example, for \(\displaystyle \cos \left( \frac{\theta }{2} \right)\), we’ll only get one solution instead of the normal two. And always check for extraneous solutions. Note that when we multiply or divide to get the variable by itself, we have to do the same with the “\(+2\pi k\)” or “\(+\pi k\)”.
Here are some problems: solve the following trig equations for 1) General Solutions , and 2) Solutions between \(\left[ {0,2\pi } \right)\) or \(\left[ {0,360{}^\circ } \right)\):
Factoring to Solve Trigonometric Equations
Note that sometimes we have to factor the equations to get the solutions, typically if they are trig quadratic equations . Then we set all factors to 0 to solve, making sure we test the answers to see if they work. We learned how to factor Quadratic Equations in the Solving Quadratics by Factoring and Completing the Square section.
Here are some general hints when solving advanced trig equations:
- When solving , simplify with identities first, if you can.
- You can square each side, but don’t divide both sides by factors with variables, since you might be missing out on solutions. If you need to cross-multiply, or multiply both sides by what’s in a denominator (even when one side equals 0 ), make sure you’re not missing solutions. (It might be a good idea to see how many solutions there are in a graphing calculator if you can). And always check for extraneous solutions : solutions must work in the original equations , and denominators can’t be 0 .
- If you get answers for any trig function that has asymptotes (like tan ), check for extraneous solutions (solutions that would be asymptotes).
Here are some examples, both solving on the interval 0 to \(2\pi \) (or \(360{}^\circ \)) and over the reals. In the last problem, the answer (\(\displaystyle \theta = \frac{{\pi k}}{2}\)) has to be “thrown out”, because of our domain restriction for cot (it falls on an asymptote); this is an extraneous solution:
Solving Trigonometric Equations Using a Calculator
We already used a calculator to find inverse trig functions here in the Inverse Trigonometric Functions section . When solving trig equations , however, it’s a little more complicated, since typically we’ll have multiple solutions. We can use a scientific calculator, or graph the functions and find intersections with a graphing calculator (usually easier). Don’t forget to change to the appropriate mode (radians or degrees) using DRG on a TI scientific calculator, or mode on a TI graphing calculator.
If just using a scientific calculator , here are some rules for solving trig problems in the intervals \(\left[ {0,2\pi } \right)\) or \(\left( {-\infty ,\infty } \right)\) in radians (substitute 180° for if using degrees). Remember these rules, which make sense if you look at the trig functions on the Unit Circle . Remember that \(k\) is any integer, negative, 0 , or positive.
- For \(\displaystyle \sin \theta =A,\,\,\theta ={{\sin }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)\) and also \(\displaystyle \theta =\pi -{{\sin }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)\). (For \(\csc \theta =A\), use \(\displaystyle {{\sin }^{{-1}}}\left( {\frac{1}{A}} \right)\)). For example, in the interval \(\left[ {0,2\pi } \right)\), for \(\displaystyle \sin \theta =.5,\,\,\theta ={{\sin }^{{-1}}}\left( {.5\,} \right)\approx .524\,\,\left( {\frac{{\pi }}{6}\,} \right)\), and also \(\displaystyle \theta =\pi -{{\sin }^{{-1}}}\left( {.5\,} \right)\approx 2.618\,\,\left( {\frac{{5\pi }}{6}\,} \right)\).
- For \(\displaystyle \cos \theta =A,\,\theta ={{\cos }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)\), and also \(\displaystyle \theta =2\pi -{{\cos }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)\), which is the same as \(\displaystyle -{{\cos }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)\). (For \(\sec \theta =A\) , use \(\displaystyle {{\cos }^{{-1}}}\left( {\frac{1}{A}} \right)\) ). For example, in the interval \(\left[ {0,2\pi } \right)\), for \(\displaystyle \cos \theta =.5,\,\,\theta ={{\cos }^{{-1}}}\left( {.5} \right)\,\approx 1.047\,\,\left( {\frac{\pi }{3}} \right)\), and also \(\displaystyle \theta =2\pi -{{\cos }^{{-1}}}\left( {.5} \right)\,\approx 5.236\,\,\left( {\frac{{5\pi }}{3}} \right)\).
- For \(\displaystyle \tan \theta =A,\,\,\theta ={{\tan }^{{-1}}}A\,\,\left( {+\,\pi k} \right)\); this will find all the solutions. (For \(\cot \theta =A\), use \(\displaystyle {{\tan }^{{-1}}}\left( {\frac{1}{A}} \right)\), but be careful with the angles \(\displaystyle \frac{\pi }{2}+\pi k\), since the calculator shows undefined, instead of 0 ; you can use \(\displaystyle \cot =\frac{{\cos }}{{\sin }}\) instead). For example, in the interval \(\left[ {0,2\pi } \right)\), for \(\displaystyle \tan \theta =1,\,\,\theta ={{\tan }^{{-1}}}\left( 1 \right)\,\approx .785\,\,\left( {\frac{\pi }{4}} \right)\), and also \(\displaystyle \theta ={{\tan }^{{-1}}}\left( 1 \right)+\pi \,\approx 3.927\,\,\left( {\frac{{5\pi }}{4}} \right)\).
Remember that when the coefficient of the argument of the inverse trig functions isn’t 1 , we need to divide the \(+\,2\pi k\) or \(+\,\pi k\) by this coefficient, since the period changes (see examples below). Typically, the default mode is radian mode , unless problem says “degrees” .
If you have access to a graphing calculator, it’s usually easier to solve trig equations. We can put the left-hand part of the equation in \({{Y}_{1}}\), the right-hand part of the equation in \({{Y}_{2}}\), and solve for the intersection(s) between 0 and \(2\pi \), or whatever the period when finding general solutions. (Use the trace feature and arrow keys to get close to each intersection, and then use the intersect feature ( 2 nd trace, 5 , enter , enter , enter ) to find the intersection.) For the reciprocal functions, take the reciprocal of what’s on the right-hand side, and use the regular trig functions.
For intervals of \(\left[ {0,2\pi } \right)\), use Xmin = 0 , and Xmax = \(2\pi \). For general solutions (over the reals), use Xmin = 0 and Xmax = the period (such as \(\displaystyle \frac{2\pi }{5}\) when you have \(\sin \left( {5x} \right)\), for example) for general solutions. Then, for your answer, add the appropriate factors of \(\pi k\), \(2\pi k\), or whatever the period of the function is.
Also remember that \({{\left( \cos \theta \right)}^{2}}\) is written as \({{\cos }^{2}}\theta \), and we can put it in the graphing calculator as \(\boldsymbol{\cos {{\left( x \right)}^{2}}}\) or \(\boldsymbol {{{\left( {\cos \left( x \right)} \right)}^{2}}}\).
Here are some examples using both types of calculators:
Solving Trig Systems of Equations
Systems of equations are needed when solving for more than one variable in equations. We learned how to solve systems of equations here in the the Systems of Linear Equations and Word Problems section , and systems of more complicated equations here in the Systems of non-Linear Equations section . Again, use either Substitution or Elimination , depending on what’s easier. Once we get the initial solution(s), we’ll can plug in a variable to get the other variable.
Here are some examples of Solving Systems with Trig Equations; solve over the reals :
Solving Trig Inequalities
Sometimes you might be asked to solve a Trig Inequality . (Links to other types of Inequalities are found here ).
We can either solve these inequalities graphically or algebraically ; let’s try one of each. Note that you can also solve these on your graphing calculator , using the Intersect feature, and then see where the inequalities “work”:
Practice these problems, and practice, practice, practice!
Hit Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.
For Practice : Use the Mathway widget below to try a Trig Solving problem. Click on Submit (the blue arrow to the right of the problem) and click on Solve for x to see the answer.
You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.
If you click on Tap to view steps , or Click Here , you can register at Mathway for a free trial , and then upgrade to a paid subscription at any time (to get any type of math problem solved!).
On to Trigonometric Identities – you’re ready!
Solve Trigonometric Equations Using the Unit Circle - Expii
Solving Trigonometric Equations Part III Lesson
- Learn how to solve trigonometric equations using square roots
- Learn how to solve trigonometric equations using squaring
- Learn how to solve trigonometric equations using identities
How to Solve Trigonometric Equations Using Square Roots, Squaring, & Identities
Unit circle, solving trigonometric equations by taking square roots, solving trigonometric equations using trigonometric identities, solving trigonometric equations using squaring, skills check:.
Solve each equation for 0 ≤ θ < 2π $$3\text{sec}^2 θ=4$$
Please choose the best answer.
Solve each equation for 0 ≤ θ < 2π $$\text{csc}\hspace{.1em}θ + \text{cot}\hspace{.1em}θ + 4=5$$
Solve each equation for 0 ≤ θ < 2π $$-2 + 4\text{cos}\hspace{.1em}θ + \text{sin}^2 θ=3 \text{cos}^2 θ$$
Congrats, Your Score is 100 %
Better Luck Next Time, Your Score is %
Ready for more? Watch the Step by Step Video Lesson | Take the Practice Test
Unit Circle
A unit circle from the name itself defines a circle of unit radius. A circle is a closed geometric figure without any sides or angles. The unit circle has all the properties of a circle, and its equation is also derived from the equation of a circle. Further, a unit circle is useful to derive the standard angle values of all the trigonometric ratios.
Here we shall learn the equation of the unit circle, and understand how to represent each of the points on the circumference of the unit circle, with the help of trigonometric ratios of cosθ and sinθ.
What is Unit Circle?
A unit circle is a circle with a radius measuring 1 unit. The unit circle is generally represented in the cartesian coordinate plane . The unit circle is algebraically represented using the second-degree equation with two variables x and y. The unit circle has applications in trigonometry and is helpful to find the values of the trigonometric ratios sine, cosine, tangent.
Unit Circle Definition
The locus of a point which is at a distance of one unit from a fixed point is called a unit circle.
Equation of a Unit Circle
The general equation of a circle is (x - a) 2 + (y - b) 2 = r 2 , which represents a circle having the center (a, b) and the radius r. This equation of a circle is simplified to represent the equation of a unit circle. A unit circle is formed with its center at the point(0, 0), which is the origin of the coordinate axes. and a radius of 1 unit. Hence the equation of the unit circle is (x - 0) 2 + (y - 0) 2 = 1 2 . This is simplified to obtain the equation of a unit circle.
Equation of a Unit Circle: x 2 + y 2 = 1
Here for the unit circle, the center lies at (0,0) and the radius is 1 unit. The above equation satisfies all the points lying on the circle across the four quadrants.
Finding Trigonometric Functions Using a Unit Circle
We can calculate the trigonometric functions of sine, cosine, and tangent using a unit circle. Let us apply the Pythagoras theorem in a unit circle to understand the trigonometric functions. Consider a right triangle placed in a unit circle in the cartesian coordinate plane. The radius of the circle represents the hypotenuse of the right triangle. The radius vector makes an angle θ with the positive x-axis and the coordinates of the endpoint of the radius vector is (x, y). Here the values of x and y are the lengths of the base and the altitude of the right triangle. Now we have a right angle triangle with the sides 1, x, y. Applying this in trigonometry , we can find the values of the trigonometric ratio, as follows:
- sinθ = Altitude/Hypoteuse = y/1
- cosθ = Base/Hypotenuse = x/1
We now have sinθ = y, cosθ = x, and using this we now have tanθ = y/x. Similarly, we can obtain the values of the other trigonometric ratios using the right-angled triangle within the unit circle. Also by changing the θ values we can obtain the principal values of these trigonometric ratios.
Unit Circle with Sin Cos and Tan
Any point on the unit circle has coordinates(x, y), which are equal to the trigonometric identities of (cosθ, sinθ). For any values of θ made by the radius line with the positive x-axis, the coordinates of the endpoint of the radius represent the cosine and the sine of the θ values. Here we have cosθ = x, and sinθ = y, and these values are helpful to compute the other trigonometric ratio values. Applying this further we have tanθ = sinθ/cosθ or tanθ = y/x.
Another important point to be understood is that the sinθ and cosθ values always lie between 1 and -1, and the radius value is 1, and it has a value of -1 on the negative x-axis. The entire circle represents a complete angle of 360º and the four quadrant lines of the circle make angles of 90º, 180º, 270º, 360º(0º). At 90º and at 270º the cosθ value is equal to 0 and hence the tan values at these angles are undefined.
Example: Find the value of tan 45º using sin and cos values from the unit circle.
We know that, tan 45° = sin 45°/cos 45°
Using the unit circle chart: sin 45° = 1/√2 cos 45° = 1/√2
Therefore, tan 45° = sin 45°/cos 45° = (1/√2)/(1/√2) = 1
Answer: Therefore, tan 45° = 1
Unit Circle Chart in Radians
The unit circle represents a complete angle of 2π radians. And the unit circle is divided into four quadrants at angles of π/2, π. 3π/2, and 2π respectively. Further within the first quadrant at the angles of 0, π/6, π/4, π/3, π/2 are the standard values, which are applicable to the trigonometric ratios. The points on the unit circle for these angles represent the standard angle values of the cosine and sine ratios. On close observation of the below figure the values are repeated across the four quadrants, but with a change in sign. This change in sign is because of the reference x-axis and y-axis, which are positive on one side and negative on the other side of the origin. Now with the help of this, we can easily find the trigonometric ratio values of standard angles, across the four quadrants of the unit circle.
Unit Circle and Trigonometric Identities
The unit circle identities of sine, cosecant, and tangent can be further used to obtain the other trigonometric identities such as cotangent, secant, and cosecant. The unit circle identities such as cosecant, secant, cotangent are the respective reciprocal of the sine, cosine, tangent. Further, we can obtain the value of tanθ by dividing sinθ with cosθ, and we can obtain the value of cotθ by dividing cosθ with sinθ.
For a right triangle placed in a unit circle in the cartesian coordinate plane, with hypotenuse, base, and altitude measuring 1, x, y units respectively, the unit circle identities can be given as,
- sinθ = y/1
- cosθ = x/1
- tanθ = sinθ/cosθ = y/x
- sec(θ = 1/x
- csc(θ) = 1/y
- cot(θ) = cosθ/sinθ = x/y
Unit Circle Pythagorean Identities
The three important Pythagorean identities of trigonometric ratios can be easily understood and proved with the unit circle. The Pythagoras theorem states that in a right-angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides. The three Pythagorean identities in trigonometry are as follows.
- sin 2 θ + cos 2 θ = 1
- 1 + tan 2 θ = sec 2 θ
- 1 + cot 2 θ = cosec 2 θ
Here we shall try to prove the first identity with the help of the Pythagoras theorem. Let us take x and y as the legs of the right-angled triangle having a hypotenuse 1 unit. Applying Pythagoras theorem we have x 2 + y 2 = 1 which represents the equation of a unit circle. Also in a unit circle, we have, x = cosθ, and y = sinθ, and applying this in the above statement of the Pythagoras theorem, we have, cos 2 θ + sin 2 θ = 1. Thus we have successfully proved the first identity using the Pythagoras theorem. Further within the unit circle, we can also prove the other two Pythagorean identities.
Unit Circle and Trigonometric Values
The various trigonometric identities and their principal angle values can be calculated through the use of a unit circle. In the unit circle, we have cosine as the x-coordinate and sine as the y-coordinate. Let us now find their respective values for θ = 0°, and θ = 90º.
For θ = 0°, the x-coordinate is 1 and the y-coordinate is 0. Therefore, we have cos0º = 1, and sin0º = 0. Let us look at another angle of 90º. Here the value of cos90º = 1, and sin90º = 1. Further, let us use this unit circle and find the important trigonometric function values of θ such as 30º, 45º, 60º. Also, we can also measure these θ values in radians . We know that 360° = 2π radians. We can now convert the angular measures to radian measures and express them in terms of the radians.
Unit Circle Table:
The unit circle table is used to list the coordinates of the points on the unit circle that corresond to common angles with the help of trigonometric ratios.
We can find the secant, cosecant, and cotangent functions also using these formulas:
- secθ = 1/cosθ
- cosecθ = 1/sinθ
- cotθ = 1/tanθ
We have discussed the unit circle for the first quadrant. Similarly, we can extend and find the radians for all the unit circle quadrants. The numbers 1/2, 1/√2, √3/2, 0, 1 repeat along with the sign in all 4 quadrants.
Unit Circle in Complex Plane
A unit circle consists of all complex numbers of absolute value as 1. Therefore, it has the equation of |z| = 1. Any complex number z = x + \(i\)y will lie on the unit circle with equation given as x 2 + y 2 = 1.
The unit circle can be considered as unit complex numbers in a complex plane, i.e., the set of complex numbers z given by the form,
z = e \(i\)t = cos t + \(i\) sin t = cis(t)
The relation given above represents Euler's formula .
Unit Circle Examples
Example 1: Does the point P (1/2, 1/2) lie on the unit circle?
We know that equation of a unit circle is:
x 2 + y 2 = 1
Substituting x = 1/2 and y = 1/2, we get:
= x 2 + y 2 = (1/2) 2 + (1/2) 2 = 1/4 + 1/4 = 1/2
Since, x 2 + y 2 ≠ 1, the point P (1/2, 1/2) does not lie on the unit circle.
Answer: Therefore (1/2, 1/2) doesn't lie on the unit circle.
Example 2: Find the exact value of tan 210° using the unit circle.
We know that
tan 210° = sin 210°/cos210°
Using the unit circle chart:
sin 210° = -1/2
cos 210° = -√3/2
tan 210° = sin 210°/cos 210°
= (-1/2)/(-√3/2)
= 1/√3
Answer: Therefore, tan 210° = 1/√3
Example 3: Find the value of sin 900° using unit circle chart.
Since, the unit circle has 0°- 360°, let us represent 900° in terms of 360°.
900° is 2 full rotations of 360° and an additional rotation of 180°.
Hence, 900° will have the same trigonometric ratio as 180°.
sin 900° = sin 180°
From the unit circle chart, we know that:
sin 180° = 0
Answer: sin 900° = 0
go to slide go to slide go to slide
Book a Free Trial Class
Practice Questions on Unit Circle
go to slide go to slide
FAQs on Unit Circle
What is unit circle in math.
A unit circle is a circle with a radius of one unit. Generally, a unit circle is represented in the coordinate plane with its center at the origin. The equation of the unit circle of radius one unit and having the center at (0, 0) is x 2 + y 2 = 1. Further, the unit circle has applications in trigonometry and is used to find the principal values of sine and cosine trigonometric ratios .
How Do you Find Sin and Cos Using the Unit Circle?
The unit circle can be used to find the values of sinθ and Cosθ. In a unit circle of radius 1 unit and having the center at (0, 0), let us take a radius inclined to the positive x-axis at an angle θ, and the endpoint of the radius as (x, y). Draw a perpendicular from the end of the radius to the x-axis and it forms a right-angled triangle with the radius as the hypotenuse. The adjacent side of this triangle is the x value, the opposite side of the triangle is the y value and the hypotenuse is of 1 unit. Further using the trigonometric ratio formula we have sinθ = Opp/Hyp = y/1, and cosθ = Adj/Hyp = x/1. Thus we have sinθ = y, and cosθ = x.
What is the Unit Circle Definition of Trig Functions?
The trigonometric function can be calculated for the principal values using the unit circle. For a unit circle having the center at the origin(0, 0), the radius of 1 unit, if the radius is inclined at an angle θ and the endpoint of the radius vector is (x, y), then cosθ = x and sinθ = y. Further, all the other trigonometric ratios can be calculated from these two values. Also, the principal values can be computed by changing the θ value.
How to Find Terminal Point on Unit Circle?
The terminal point on a unit circle can be found with the help of the equation of the unit circle x 2 + y 2 = 1. If the given point satisfies this equation then it is a point lying on the unit circle. Further, the terminal point on the unit value can be found for the θ value, by finding the values of cosθ and sinθ.
What is the Equation of Unit Circle?
The equation of a unit circle is x 2 + y 2 = 1. Here it is considered that the unit circle has its center at the origin(0, 0) of the coordinate axes, and has a radius of 1 unit. This equation of unit circle has been derived using the help of the distance formula.
How Do you Derive the Equation of a Unit Circle?
The equation of a unit circle can be calculated using the distance formula of coordinate geometry. For a circle having the center at the origin(0, 0), the radius of 1 unit, any point on the circle can be taken as (x, y). Applying the definition of a circle, and using the distance formula we have (x - 0) 2 + (y - 0) 2 = 1, which can be simplified as x 2 + y 2 = 1.
When is Tan Undefined on the Unit Circle?
The unit circle having an equation of x 2 + y 2 = 1 is helpful to find the trigonometric ratios of sinθ = y and cosθ = x. Using these values we can conveniently find the value of tanθ = sinθ/cosθ = y/x. Tanθ will be undefined for cosθ = 0, i.e., when θ is equal to 90° and 270°.
What is the Connection Between Right Triangles and the Unit Circle?
The right triangles and a unit circle are uniquely connected. Any point on the unit circle can be visualized as a right triangle with radius as the hypotenuse of the right triangle and the coordinates of the point as the other two sides of the right triangle. The equation of a circle x 2 + y 2 = 1 completely satisfies the Pythagoras theorem related to the right triangle. Also, the right triangle within the unit circle is helpful to derive the trigonometric ratio values.
What is the Unit Circle Used for?
The unit circle is prominently useful in trigonometry. For the trigonometric ratios of sinθ, cosθ, tanθ, their principal angle values of 0º, 30º, 45º, 60º, 90º can be easily calculated using the unit circle. Additionally, the unit circle is useful to represent complex numbers in the argand plane.
What are the Quadrants of the Unit Circle?
The unit circle has four quadrants similar to the quadrants in the coordinate system. The four quadrants are of equal area and they represent one-fourth of the area of the circle . Each of the quadrants subtends an angle of 90º or a right angle at the center of the circle.
How Do you Describe a Unit Circle in Terms of Complex Numbers?
A unit circle consists of all complex numbers of absolute value as 1, thus any complex number z = x + \(i\)y will lie on the unit circle with equation given as x 2 + y 2 = 1. Therefore, the equation of the unit circle can be given as |z| = 1.
IMAGES
VIDEO
COMMENTS
The four steps for solving an equation include the combination of like terms, the isolation of terms containing variables, the isolation of the variable and the substitution of the answer into the original equation to check the answer.
Whether you love math or suffer through every single problem, there are plenty of resources to help you solve math equations. Skip the tutor and log on to load these awesome websites for a fantastic free equation solver or simply to find an...
The periods of the trigonometric functions sine and cosine are both 2 times pi. The functions tangent and cotangent both have a period of pi. The general formula for the period of a trigonometric function can be determined by dividing the r...
This video explains how to solve a basic trigonometric equation with solutions that can be found using the unit circle and reference
http://www.mathpowerline.com Solving trig equations is made a lot easier if you know how to use the unit circle. These problems are when the
A simple way to visualise trigonometric equations lets us solve them quickly using spatial intuition. No need for CAST or ASTC (all stations
Solving Trigonometric Equations with the Unit Circle ... 10K views 5 years ago Trigonometry. 10,233 views • Nov 25, 2017 • Trigonometry.
Solving Trigonometric Equations Using the Unit Circle · \displaystyle 2{{\cos }^{2}}\theta =1 · \displaystyle \sin \left( {\theta +\frac{\pi }{{18}}} \right)=0 · \
The simplest way to solve a trig equation like sin(x) = 0.5 is just to use the unit circle.
Using the unit circle diagram, draw a line “tangent” to the unit circle where the hypotenuse contacts the unit circle. This line is at right angles to the
Starting from ( 1 , 0 ) (1,0) (1,0)left parenthesis, 1, comma, 0, right parenthesis, move along the unit circle in the counterclockwise direction until the
Solving Linear Trig Equations(Unit Circle). When solving a linear trig equation it is important to read the directions. Are you working in degrees or.
How to Solve Trigonometric Equations Using Square Roots, Squaring, & Identities · Unit Circle · Solving Trigonometric Equations by Taking Square Roots · Solving
The general equation of a circle is (x - a)2 + (y - b)2 = r2, which represents a circle having the center (a, b) and the radius r.