6. Trigonometric Identities and More Equations

Identify the basic trigonometric function graphed, and determine whether it is even or odd.

<IMAGE>

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

tan 72°

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

-tan x cos x

II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

Use the given information to find each of the following.

sin x, given cos 2x = 2/3 , with π < x < 3π/2

In Exercises 67–74, rewrite each expression in terms of the given function or functions. $\frac{\tan x+\cot x}{\csc x}$; $\cos x$

In the context of the $unit circle$, which of the following best represents an $intercepted arc$?

Verify that each equation is an identity.

sin² θ (1 + cot² θ) - 1 = 0

Find the remaining five trigonometric functions of θ.

cos θ = 1/5, θ in quadrant I

Verify that each equation is an identity.

(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ

Factor each trigonometric expression.

cot⁴ x + 3 cot² x + 2

Perform each indicated operation and simplify the result so that there are no quotients.

(tan x + cot x)²

Which of the following correctly expresses the relationship between the length of an arc $s$ of a circle, its radius $r$, and the central angle $\theta$ in radians?

Identify the most helpful first step in verifying the identity.

$\(\sec\)^3\(\theta\)=\(\sec\]\theta\)+\(\frac{\tan^2\theta}{\cos\theta}\)$

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

cot t tan t