Trigonometric Equations

Solving for the angle

(The first of two note days and a work day)

(6.2)(1)

POD

Solve for angles in one rotation, then for a general solution.

2

1sin

Trig Equations

What makes it an equation?

What makes it a trig equation?

We’re going to use a lot of inverse trig functions today.

Note: Unless otherwise specified, operate in radians.

Trig Equations

Steps to solve them:

1. Solve the equation for sinθ, cosθ, or tanθ.2. Find values of θ that satisfy the equation in one

rotation.3. Consider all possible values of θ for a general

solution.4. If needed, undo any substitutions and solve for any

variables.

We’ve done some of this already when we used inverse trig functions, say, in the POD.

Use it

A riff on the POD.

Step 1 is done.

Step 2 Solve for 0 ≤ θ ≤ 2π.

Step 3 Solve for all θ.

2

1sin

Use it

A riff on the POD.

Step 2 Solve for 0 ≤ θ ≤ 2π.

θ = 7π/6 and 11π/6

Step 3 Solve for all θ.

θ = 7π/6 ± 2πn and 11π/6 ± 2πn

2

1sin

Use it

A step beyond– solve for the angle, then for the variable (step 4 in the method).

In this case, find the general solution and then give all values of x in the interval 0 ≤ x ≤ 2π.

02cos x

Use it

A step beyond– solve for the angle, then for the variable (step 4 in the method).

First step is done.

Second step, solve for 1 rotation, then a general solution.

02cos x

Use it

Second step, solve for 1 rotation to build a general solution.

In one rotation: θ = π/2 and θ = 3π/2.

(Notice how I substituted θ for 2x; it’s easier to work with.)

Third step, general solution:

θ = π/2 ± 2πn and θ = 3π/2 ± 2πn

Combined general solution: π/2 + πn

02cos x

Use it

Combined general solution: θ = π/2 + πn

Final step, remove the substitution and solve for x.

02cos x

Use it

Combined general solution: θ = π/2 + πn

From the general solution

θ = 2x = π/2 ± πn

x = π/4 ± πn/2

So, in the interval 0 ≤ x ≤ 2π, x = π/4, 3π/4, 5π/4, 7π/4.

02cos x

Use it

Combined general solution: θ = π/2 + πn

x = π/4 ± πn/2

Check: Compare the graph of y = cos x to y = cos 2x. What changes? What are the x-intercepts?

How does this graph relate to our solution?

Use it

Incorporate factoring to solve for sin θ and tan θ.

What should you NOT do?

xxx sintansin

Use it

Incorporate factoring to solve for sin θ and tan θ.

Now, solve for the angles.

1tan

0sin

0)1(tansin

0sintansin

sintansin

x

x

xx

xxx

xxx

Use it

Incorporate factoring to solve for sin θ and tan θ.

tan θ = 1 sin θ = 0

One rotation θ = π/4, 5π/4 θ = 0, π

General sol. θ = π/4 ± πn θ = ±πn

This means that any angle in either category will make the equation true. Test it with θ = π and π/4.

1tan

0sin

x

x

Use it

Remember the trig identities. Factor again.

01cossin2 2 tt

Use it

Remember the trig identities. Factor again.

1cos2

1cos

0)1)(cos1cos2(

01coscos2

01coscos2

01coscos22

01cos)cos1(2

01cossin2

2

2

2

2

2

t

t

tt

tt

tt

tt

tt

tt

Use it

Now solve for t.

1cos2

1cos

t

t

Use it

Now solve for t.

cos t = ½ cos t = -1

One rotation t = π/3, 5π/3 t = π

General sol. t = π/3 ± 2πn t = π ± 2πn

t = 5π/3 ±2πn

(Combined: t = ±π/3 ± 2πn)

1cos2

1cos

t

t