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Problem 5.56c
Use the given information to find the quadrant of s + t. See Example 3.
cos s = -1/5 and sin t = 3/5, s and t in quadrant II
Problem 5.58
Write each expression as a sum or difference of trigonometric functions. See Example 7.
2 cos 85° sin 140°
Problem 5.58
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 sec θ
Problem 5.58
Verify that each equation is an identity.
(sec α - tan α)² = (1 - sin α)/(1 + sin α)
Problem 5.58
Verify that each equation is an identity.
csc A sin 2A - sec A = cos 2A sec A
Problem 5.6
Match each expression in Column I with its value in Column II.
(2 tan (π/3))/(1 - tan² (π/3))
Problem 5.60
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.
cot² θ(1 + tan² θ)
Problem 5.6
For each expression in Column I, choose the expression from Column II that completes an identity.
6. sec² x = ____
Problem 5.6
Match each expression in Column I with its equivalent expression in Column II.
sin 60° cos 45° - cos 60° sin 45°
Problem 5.60
Verify that each equation is an identity.
[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ
Problem 5.60
Verify that each equation is an identity.
2 cos² θ - 1 = (1 - tan² θ)/(1 + tan² θ)
Problem 5.62
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.
(sec θ - 1) (sec θ + 1)
Problem 5.62
Verify that each equation is an identity.
sin(x + y) + sin(x - y) = 2 sin x cos y
Problem 5.62
Verify that each equation is an identity.
1/(sec α - tan α) = sec α + tan α
Problem 5.62
Verify that each equation is an identity.
sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)
Problem 5.64
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)
Problem 5.64
Verify that each equation is an identity. See Example 4.
tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x tan y)
Problem 5.64
Verify that each equation is an identity.
(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ
Problem 5.64
Verify that each equation is an identity.
sin³ θ = sin θ - cos² θ sin θ
Problem 5.65
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.
cos 18°
Problem 5.66
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 - sin²(-θ)]/[1 + cot²(-θ)]
Problem 5.66
Verify that each equation is an identity.
sin(s + t)/cos s cot t = tan s + tan t
Problem 5.66
Verify that each equation is an identity.
sin² θ (1 + cot² θ) - 1 = 0
Problem 5.66
Verify that each equation is an identity.
2 cos² (x/2) tan x = tan x+ sin x
Problem 5.67
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.
cot 18°
Problem 5.67
Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos( π/2 + x) = -sin x
Problem 5.68
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 5.68
Verify that each equation is an identity.
sin(x + y)/cos(x - y) = (cot x + cot y)/(1 + cot x cot y)
Problem 5.68
Verify that each equation is an identity.
(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1
Problem 5.68
Verify that each equation is an identity.
(1/2)cot (x/2) - (1/2) tan (x/2) = cot x