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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 5.69
Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = cos² x - sin² x
Problem 5.70
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 5.70
Verify that each equation is an identity.
(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α
Problem 5.70
Verify that each equation is an identity.
(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t
Problem 5.70
Verify that each equation is an identity.
(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))
Problem 5.71
Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = 1 - 2 sin² x
Problem 5.72
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.
cos θ (cos θ - sec θ)
Problem 5.72
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°
Problem 5.72
Verify that each equation is an identity.
tan² α sin² α = tan² α + cos² α - 1
Problem 5.73
Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = (cot² x - 1)/(cot² x + 1)
Problem 5.74
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.
(sec²θ - 1)/(csc²θ - 1)
Problem 5.74
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 72°
Problem 5.74
Verify that each equation is an identity.
sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)
Problem 5.76
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.
tan(-θ)/sec θ
Problem 5.76
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.
sin 162°
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 5.78
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.
-sec² (-θ) + sin² (-θ) + cos² (-θ)
Problem 5.78
Verify that each equation is an identity.
sin θ + cos θ = sin θ/(1 - cot θ) + cos θ/(1 - tan θ)
Problem 5.8
Find values of the sine and cosine functions for each angle measure.
2θ, given cos θ = -12/13 and sin θ > 0
Problem 5.80
Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.
Problem 5.8
Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
cot(-θ)/sec(-θ)
Problem 5.8
Match each expression in Column I with its value in Column II.
8. tan (-π/8)
Problem 5.8
Match each expression in Column I with its equivalent expression in Column II.
(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))
Problem 5.80
Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
Problem 5.81
Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
x + y = 9, 2x + y = -1
Problem 5.82
Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
5x - 2y + 4 = 0, 3x + 5y = 6
Problem 5.82
Verify that each equation is an identity.
(sec α + csc α) (cos α - sin α) = cot α - tan α
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 5.86
Verify that each equation is an identity.
sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x