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Problem 5.48
Verify that each equation is an identity.
(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)
Problem 5.48
Verify that each equation is an identity.
(tan² α + 1)/ sec α = sec α
Problem 5.50
Express each function as a trigonometric function of x. See Example 5.
cos 3x
Problem 5.50
Perform each transformation. See Example 2.
Write cot x in terms of csc x.
Problem 5.50
Verify that each equation is an identity.
tan (θ/2) = csc θ - cot θ
Problem 5.50
Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.
cos(270° + θ)
Problem 5.50
Verify that each equation is an identity.
sin² β (1 + cot² β) = 1
Problem 5.50
Verify that each equation is an identity.
2 cos³ x - cos x = (cos² x - sin² x)/sec x
Problem 5.52
Perform each transformation. See Example 2.
Write sec x in terms of sin x.
Problem 5.52
Express each function as a trigonometric function of x. See Example 5.
cos 4x
Problem 5.52
Verify that each equation is an identity.
cos x = (1 - tan² (x/2))/(1 + tan² (x/2))
Problem 5.52
Find cos(s + t) and cos(s - t).
cos s = - 8/17 and cos t = - 3/5, s and t in quadrant III
Problem 5.52
Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α
Problem 5.52
Verify that each equation is an identity.
(sin 2x)/(sin x) = 2/sec x
Problem 5.52a
Use the given information to find sin(s + t). See Example 3.
sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III
Problem 5.52b
Use the given information to find tan(s + t). See Example 3.
sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III
Problem 5.52c
Use the given information to find the quadrant of s + t. See Example 3.
sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III
Problem 5.54
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 cos θ
Problem 5.54
Find cos(s + t) and cos(s - t).
sin s = 2/3 and sin t = -1/3, s in quadrant II and t in quadrant IV
Problem 5.54
Verify that each equation is an identity.
(sin² θ)/cos θ = sec θ - cos θ
Problem 5.54
Verify that each equation is an identity.
(2 tan B)/(sin 2B) = sec² B
Problem 5.54a
Use the given information to find sin(s + t). See Example 3.
cos s = -15/17 and sin t = 4/5, s in quadrant II and t in quadrant I
Problem 5.54b
Use the given information to find tan(s + t). See Example 3.
cos s = -15/17 and sin t = 4/5, s in quadrant II and t in quadrant I
Problem 5.54c
Use the given information to find the quadrant of s + t. See Example 3.
cos s = - 15/17 and sin t = 4/5, s in quadrant II and t in quadrant I
Problem 5.56
Find cos(s + t) and cos(s - t).
cos s = √2/4 and sin t = - √5/6, s and t in quadrant IV
Problem 5.56
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 5.56
Verify that each equation is an identity.
sec⁴ x - sec² x = tan⁴ x + tan² x
Problem 5.56
Verify that each equation is an identity.
(2 cot x)/(tan 2x) = csc² x - 2
Problem 5.56a
Use the given information to find sin(s + t). See Example 3.
cos s = -1/5 and sin t = 3/5, s and t in quadrant II
Problem 5.56b
Use the given information to find tan(s + t). See Example 3.
cos s = -1/5 and sin t = 3/5, s and t in quadrant II