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## Solving Trigonometric Equations

## Solving Trigonometric Equations Using the Unit Circle

## Solving Trigonometric Equations – General Solutions

## Solving Trigonometric Equations with Multiple Angles

## Factoring to Solve Trigonometric Equations

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).

## Solving Trigonometric Equations Using a Calculator

- 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 \(\displaystyle \theta ={{\tan }^{{-1}}}\left( 1 \right)+\pi \,\approx 3.927\,\,\left( {\frac{{5\pi }}{4}} \right)\).

Here are some examples using both types of calculators:

## Solving Trig Systems of Equations

Here are some examples of Solving Systems with Trig Equations; solve over the reals :

## Solving Trig Inequalities

Practice these problems, and practice, practice, practice!

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

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 $$\text{csc}\hspace{.1em}θ + \text{cot}\hspace{.1em}θ + 4=5$$

Better Luck Next Time, Your Score is %

Ready for more? Watch the Step by Step Video Lesson | Take the Practice Test

## Unit Circle

## What is Unit Circle?

## 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

Equation of a Unit Circle: x 2 + y 2 = 1

## Finding Trigonometric Functions Using a Unit Circle

## Unit Circle with Sin Cos and Tan

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

## Unit Circle and Trigonometric Identities

## Unit Circle Pythagorean Identities

## Unit Circle and Trigonometric Values

## Unit Circle Table:

We can find the secant, cosecant, and cotangent functions also using these formulas:

## Unit Circle in Complex Plane

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:

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.

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°.

From the unit circle chart, we know that:

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## 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.