Solving General Trigonometric Equations Using Graphing

Solving general trigonometric equations is made even easier with graphing. In order to graph one of these equations you must separate the variables in relation to sine and cosine from the constants such as numbers or situations such as cos(64°). 1. You must define a period which is input into the X window range on your calculator 2. Set your Y range to -5 to 5 simply because it will usually show what you need to find. 3. Input the sine and cosine portion of your equation into your Y1 space. 4. Enter the constant side of your equation into Y2, and press 'Graph' 5. Now use the Intersect ability of your calculator to find the values for the equation (For the Ti-83 and Ti84 calculators, this fuction is found by pressing the 2nd and then Trace buttons. It is option 5). Then add the appropriate period to the end of the answer such as 180(degrees)k or 360(degrees)k and so on.

4) Plug the equation into your calculator.
Y1 = tan(x)² - 1/cos(x)

5) Use the second trace function to find intersections (For Ti-83 and Ti-84 calculators)

6) Divide the intersection's X value by Pi (This is done by pressing 2nd and Mode buttons to exit the graph, and then press 2nd + (-) keys to copy the intersection X value. Then, just divide by Pi) x = π + 2kπ
x = π/3 + 2kπ
x = 5π/3 + 2kπ

Solving Trigonometric Equations For Exact Answers (needs to be clarified)
Solving trigonometric equations for exact answers is pretty simple. First, solve the equation so the function (tanx, sinx, cosx, etc.) is alone, as demonstrated above. Then, find the value given to that function on the unit circle. For example, if the equation looked like this, cosx=1/2, you would find the cosine value of 1/2 on the unit circle and find which angle that it relates to. In this case, the angles that match are π/3, and 5π/3. Make sure you find all of the answers that apply to the value you solved for. All of the answers that you find on the unit circle should be within the interval notation. If the interval notion was [0, 2π] then both of the angles above would be correct. If it is not you can often add or subtract a rotation.

When solving Trigononmetric equations there will be a given domain, which follows interval notation.
Interval notation is basically just shown by either using brackets or parentheses. If brackets are used then this indicates that the variable is included in the domain. If parentheses are used, then this indicates that the variable is not included in the domain.

Ex) (a,b] - here the variable "a" would not be included, and "b" would be included. So the domain would be a<θ≤b.
In other words:
() = not included
[]= included

## Table of Contents

## Solving General Trigonometric Equations Using Graphing

Solving general trigonometric equations is made even easier with graphing. In order to graph one of these equations you must separate the variables in relation to sine and cosine from the constants such as numbers or situations such as cos(64°).1.You must define a period which is input into the X window range on your calculator2.Set your Y range to -5 to 5 simply because it will usually show what you need to find.3.Input the sine and cosine portion of your equation into your Y1 space.4.Enter the constant side of your equation into Y2, and press 'Graph'5.Now use the Intersect ability of your calculator to find the values for the equation (For the Ti-83 and Ti84 calculators, this fuction is found by pressing the 2nd and then Trace buttons. It is option 5). Then add the appropriate period to the end of the answer such as 180(degrees)k or 360(degrees)k and so on.Ex)

1)

Simplifytan²x - secx - 1 = 0

tan²x - secx = 1

2)

Find Periodtan²x is π/2

secx is 2π

Choose the largest of the periods

3)

Set windowXmin = 0

Xmax = 2π

Xscl = 1

Ymin = -5

Ymax = 5

Yscl = 1

Xres = 1

4)

Plug the equation into your calculator.Y1 = tan(x)² - 1/cos(x)

5)

Use the second trace function to find intersections (For Ti-83 and Ti-84 calculators)6)

Divide the intersection's X value by Pi (This is done by pressing 2nd and Mode buttons to exit the graph, and then press 2nd + (-) keys to copy the intersection X value. Then, just divide by Pi)x = π + 2kπ

x = π/3 + 2kπ

x = 5π/3 + 2kπ

(needs to be clarified)Solving Trigonometric Equations For Exact AnswersSolving trigonometric equations for exact answers is pretty simple. First, solve the equation so the function (tanx, sinx, cosx, etc.) is alone, as demonstrated above. Then, find the value given to that function on the unit circle. For example, if the equation looked like this, cosx=1/2, you would find the cosine value of 1/2 on the unit circle and find which angle that it relates to. In this case, the angles that match are π/3, and 5π/3. Make sure you find all of the answers that apply to the value you solved for. All of the answers that you find on the unit circle should be within the interval notation. If the interval notion was [0, 2π] then both of the angles above would be correct. If it is not you can often add or subtract a rotation.

1) Isolate the trig function.

ex.) cosxsinx = sinx

cosx = 1

2) Find the answer on the unit circle.

ex.) cosx = 1 at 0 and 2π

3) Make sure the answer you found fits into the interval notation.

ex.) If interval notation is (0, 2π], only 2π would be correct.

Click on link below to get tips on how to start to solve trigonometric equations.

Here is a link that shows you how to solve trig equations with a graphing calculator

When solving Trigononmetric equations there will be a given domain, which follows interval notation.

Interval notation is basically just shown by either using brackets or parentheses. If brackets are used then this indicates that the variable is included in the domain. If parentheses are used, then this indicates that the variable is not included in the domain.

Ex) (a,b] - here the variable "a" would not be included, and "b" would be included. So the domain would be a<θ≤b.

In other words:

() = not included

[]= included

Tips for Solving Trig EquationsUnit Circle will help you to figure out Tan(X) value.