Question

Determine the exact values of the following:

a) (𝑠𝑖𝑛30)(𝑠𝑖𝑛60)
b) (𝑐𝑜𝑠45) (𝑡𝑎𝑛45)
c)
(𝑐𝑜𝑠60)(𝑠𝑖𝑛30)/
𝑡𝑎𝑛30
d) (𝑠𝑖𝑛45)(𝑐𝑜𝑠60)(𝑡𝑎𝑛30)e) 𝑠𝑖𝑛
^2
60 + (𝑐𝑜𝑠30)

Answer 1

[AI-generated answer — Bot GPT 3.5]

a) Using the trigonometric identity sin(A)sin(B) = (1/2)[cos(A - B) - cos(A + B)]

sin(30)sin(60) = (1/2)[cos(30 - 60) - cos(30 + 60)]
= (1/2)[cos(-30) - cos(90)]
= (1/2)[cos(30) - 0]
= (1/2)(√3/2)
= √3/4

b) Using the trigonometric identity cos(A)tan(B) = sin(A)
cos(45)tan(45) = sin(45)

c) Using the trigonometric identity cos(A)sin(B)/tan(C) = sin(A - B)/sin(C)
cos(60)sin(30)/tan(30) = sin(60 - 30)/sin(30)
= sin(30)/sin(30)
= 1

d) sin(45)cos(60)tan(30) = [sin(45)][cos(60)][sin(30)/cos(30)]
= [sin(45)][cos(60)][2sin(30)]
= (√2/2)(1/2)(1)
= √2/4

e) Taking trigonometric identities into consideration,
sin^2(60) + cos(30) = 3/4 + √3/2