Trigonometric Equations and Inequalities

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

Trig equations such as $\sin x = c$ are solved by: (1) finding the principal solution via inverse trig, (2) using symmetry to find the second solution within one period, and (3) adding multiples of the period for the general solution. Quadratic-in-trig equations factor like ordinary quadratics — substitute $u = \sin x$ (or $\cos x$ , $\tan x$ ) and solve for $u$ first.

Solutions in [0, 2π)

\sin x = c

(

|c| \leq 1

)

x = \arcsin c \;\text{ and }\; x = \pi - \arcsin c

\cos x = c

(

|c| \leq 1

)

x = \arccos c \;\text{ and }\; x = 2\pi - \arccos c

\tan x = c

(period

\pi

)

x = \arctan c + n\pi

AP Tip: Sine has TWO solutions per period that share the same value (one in Q1/Q2, one in Q3/Q4 when negative). Cosine similarly. Tangent has only one per period.

Type 1

Solve a basic trig equation

Find the principal solution with inverse trig, then use symmetry to find the second solution.

Example 1

Find all

x \in [0, 2\pi)

with

\sin x = 1/2

. (A)

\pi/6

(B)

\pi/6, 5\pi/6

(C)

\pi/3, 2\pi/3

(D)

\pi/6, 11\pi/6

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Type 2

General solution with period

Add multiples of the period ( $2\pi$ for sin/cos, $\pi$ for tan) to each solution within one period.

Example 2

Write the general solution to

\tan x = 1

. (A)

x = \pi/4 + n\pi

(B)

x = \pi/4 + 2\pi n

(C)

x = \pi/2 + \pi n

(D)

x = \pi/4

only

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Type 3

Factor or substitute

Substitute $u = \sin x$ (or similar). Solve the resulting algebraic equation, then back-substitute.

Example 3

Find all

x \in [0, 2\pi)

with

2 \sin^2 x - \sin x - 1 = 0

. (A)

\pi/6, 5\pi/6, 3\pi/2

(B)

\pi/2, 3\pi/2

(C)

\pi/2, 7\pi/6, 11\pi/6

(D)

\pi/4, 3\pi/4, 5\pi/4

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