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Problem 4
Match each expression in Column I with its value in Column II.
cos² (π/6) - sin² (π/6)
Problem 5.54
Verify that each equation is an identity.
(sin² θ)/cos θ = sec θ - cos θ
Problem 5.30
Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2
Problem 5.1.12
Find sin θ.
cos θ = 5/6, θ in quadrant I
Problem 5.6.18
The half-angle identity
tan A/2 = ± √[(1 - cosA)/(1 + cos A)]
can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity
tan A/2 = sin A/(1 + cos A)
can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)
Problem 5.1.43
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.
cos² x
Problem 5.1.56
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
csc θ cos θ tan θ
Problem 5.1.10
Use identities to correctly complete each sentence.
If sin θ = ⅔, then -sin(-θ) = ________________.
Problem 5.84
Verify that each equation is an identity.
(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1
Problem 5.66
Verify that each equation is an identity.
sin² θ (1 + cot² θ) - 1 = 0
Problem 5.82
Verify that each equation is an identity.
(sec α + csc α) (cos α - sin α) = cot α - tan α
Problem 5.36
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
Problem 5.1.70
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
(sin θ - cos θ) (csc θ + sec θ)
Problem 5.1.30
Identify the basic trigonometric function graphed, and determine whether it is even or odd.
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Problem 5.1.22
Find sinθ.
csc θ = -8/5
Problem 5.68
Verify that each equation is an identity.
(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1
Problem 5.1.68
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
csc θ - sin θ
Problem 5.1.39
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
Problem 5.26
Factor each trigonometric expression.
(tan x + cot x)² - (tan x - cot x)²
Problem 5.42
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1/ tan² α + cot α tan α
Problem 5.24
Factor each trigonometric expression.
sec² θ - 1
Problem 5.1.50
Perform each transformation. See Example 2.
Write cot x in terms of csc x.
Problem 5.1.28
Identify the basic trigonometric function graphed, and determine whether it is even or odd.
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Problem 5.6.2
Determine whether the positive or negative square root should be selected.
cos 58° = ±√ (1 + cos 116°)/2]
Problem 5.1.32
Find the remaining five trigonometric functions of θ.
cos θ = 1/5, θ in quadrant I
Problem 5.76
Verify that each equation is an identity.
(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ
Problem 5.69
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.
csc 18°
Problem 5.40
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
csc² t - 1
Problem 5.1.38
Find the remaining five trigonometric functions of θ.
cos θ = -1/4, sin θ > 0
Problem 5.1.64
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
1 + cot(-θ)/cot(-θ)