Equations Requiring Identities

Many advanced trig equations cannot be solved directly. You must first simplify using identities to rewrite the equation in a solvable form. Common strategies include:

Using double angle formulas to reduce the equation to a single angle
Applying Pythagorean identity (sin^2 + cos^2 = 1) to convert to one trig function
Factoring after converting, treating sin or cos as a variable
Using compound angle formulas to combine or separate terms

Key Strategy: Always aim to get an equation in terms of a single trig function of a single angle before solving.

Finding Multiple Solutions in a Domain

When solving for x in a restricted domain such as 0 ≤ x ≤ 2π, you must find all solutions, not just one. Use the ASTC (All Students Take Calculus) quadrant rule:

Q2: S

sin positive

π - α

Q1: A

all positive

α

Q3: T

tan positive

π + α

Q4: C

cos positive

2π - α

General Solutions

The general solution gives all possible solutions using an integer parameter n:

sin x = a: x = nπ + (-1)^n sin^(-1)(a)

cos x = a: x = 2nπ ± cos^(-1)(a)

tan x = a: x = nπ + tan^(-1)(a)

where n is any integer (n ∈ Z). These formulas capture infinitely many solutions at once.

Key Vocabulary

Domain

The specified interval over which solutions must be found, e.g., 0 ≤ x ≤ 2π.

General Solution

A formula using an integer parameter n that generates all solutions of a trig equation.

Reference Angle

The acute angle α used with ASTC to find all solutions in [0, 2π].

ASTC Rule

A mnemonic for remembering which trig functions are positive in each quadrant: All, Sin, Tan, Cos.

Worked Examples

1

Solve 2sin^2(x) - 1 = 0 for 0 ≤ x ≤ 2π.

Step 1: sin^2(x) = 1/2, so sin(x) = ±1/√2 = ±√2/2

Step 2: Reference angle: α = π/4

Step 3: sin(x) = √2/2: x = π/4, 3π/4 (Q1 and Q2)

Step 4: sin(x) = -√2/2: x = 5π/4, 7π/4 (Q3 and Q4)

Answer: x = π/4, 3π/4, 5π/4, 7π/4

2

Solve cos 2x = cos x for 0 ≤ x ≤ 2π.

Step 1: Replace cos 2x with 2cos^2x - 1: 2cos^2x - 1 = cos x

Step 2: 2cos^2x - cos x - 1 = 0

Step 3: Factor: (2cos x + 1)(cos x - 1) = 0

Step 4: cos x = -1/2 gives x = 2π/3, 4π/3; cos x = 1 gives x = 0, 2π

Answer: x = 0, 2π/3, 4π/3, 2π

3

Find the general solution of tan x = 1.

Step 1: tan^(-1)(1) = π/4

Step 2: General solution for tan x = a is x = nπ + tan^(-1)(a)

Answer: x = nπ + π/4 where n is any integer

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

How many solutions does sin x = 1/2 have in 0 ≤ x ≤ 2π?

1 4 3 2

Question 2

To solve sin 2x = sin x, a useful first step is:

Replace sin 2x with 2 sin x cos x, then rearrange and factor Divide both sides by sin x Square both sides Use the inverse sine function

Question 3

The general solution of cos x = 0 is:

x = nπ x = 2nπ ± π/2 x = nπ + π/2 x = π/2

Question 4

In which quadrants is tan x negative?

Q1 and Q3 Q1 and Q2 Q2 and Q4 Q3 and Q4

Question 5

The equation 2cos^2x + cosx - 1 = 0 can be factored as:

(cosx + 1)(2cosx + 1) = 0 (2cosx + 1)(cosx - 1) = 0 (cosx - 1)(cosx + 2) = 0 (2cosx - 1)(cosx + 1) = 0

Advanced Trigonometric Equations