Trigonometric Equations

Definition

A trigonometric equation is an equation that contains a trig expression with a variable, such as .

Example:

A solution to this equation is because

Is the only solution to this equation?

Consider:

This graph shows 5 different solutions to the equation So how can we represent all of the solutions?

Since the period of the sine function is first find all solutions in Those solutions are and .

Any multiple of can be added to these values and the sine is still So, all solutions can be given by with n any integer.

Find 4 more solutions to

Equations involving a Single Trig FunctionTo solve an equation involving a single trig function:1. Isolate the function on one side

of the equation.2. Solve for the variable

Solve:

3sin 𝑥−2=5sin 𝑥−1

Solve:

5sin 𝑥=3sin 𝑥+√3

Equations Involving Multiple Angles

tan 3𝑥=10≤𝑥<2𝜋

Solve:

Solve:

,

Trig Equations in Quadratic FormForm: where is a trig function Solve by usual Quadratic

Methodsa. Factorb. Quadratic Formulac. Square Root Method

Solve by factoring:

2𝑐𝑜𝑠2𝑥+cos x−1=0 ,0 ≤𝑥<2𝜋

Solve by factoring

2𝑠𝑖𝑛2𝑥−3sin 𝑥+1=0 ,0≤𝑥<360 °

Solve

2𝑠𝑖𝑛2𝑥=sin𝑥+3 ,0 ≤𝑥<2𝜋

Solve

𝑠𝑖𝑛2𝑥+3 𝑠𝑖𝑛𝑥−5=0 ,0≤ 𝑥<360 °

3𝑐𝑜𝑠2 𝑥−4cos 𝑥=−4Solve

Solve

4 𝑠𝑖𝑛2𝑥−1=0 ,0≤ 𝑥<2𝜋

Solve

4𝑐𝑜𝑠2𝑥−3=0 ,0 ≤𝑥<360 °

Separate Two Functions by Factoring

𝑡𝑎𝑛𝑥 𝑠𝑖𝑛2𝑥=3 tan 𝑥 ,0≤ 𝑥<2𝜋

Separate by Factoring

𝑠𝑖𝑛𝑥𝑡𝑎𝑛𝑥=sin𝑥

Solve

𝑡𝑎𝑛2 𝑥𝑐𝑜𝑠𝑥=𝑡𝑎𝑛2𝑥 ,0 ≤𝑥<360 °

Solve

𝑐𝑜𝑡2𝑥𝑠𝑖𝑛𝑥=𝑐𝑜𝑡2 𝑥 ,0≤ 𝑥<2𝜋

Using Identities to Solve Trig Equations (All Solve:

2𝑠𝑖𝑛2𝑥−3cosx=0

𝑐𝑜𝑠2𝑥+3sin 𝑥−2=0

cos 2𝑥+sin 𝑥=0

sin 𝑥cos 𝑥=12

sin 𝑥− cos 𝑥=1

cos 𝑥−sin 𝑥=−1