Question

Use either the cofunction or reciprocal identities to complete each of the following:

a. If sin 18o=0.3
then csc 18o
b. If sin 18o=0.3
then cos 72o
c. If cot 53o=0.75
then tan 53o
d. If cot 53o=0.75
then the tangent of what angle is 0.75 ?
e. If sec 60o=2
then the cosine of what angle is 0.5 ?

Answer 1

a. Using the reciprocal identity for cosecant: csc(x) = 1/sin(x), we have:

csc(18) = 1/sin(18) = 1/0.3 = 3.333...

b. Using the cofunction identity for cosine and sine: cos(x) = sin(90 - x), we have:

cos(72) = sin(90 - 18) = sin(72) ≈ 0.9511

c. Using the reciprocal identity for tangent: tan(x) = 1/cot(x), we have:

tan(53) = 1/cot(53) = 1/0.75 = 1.333...

d. Using the tangent identity for cotangent: tan(x) = 1/cot(x), we have:

tan(x) = 1/cot(53) = 1/0.75 = 1.333...

Taking the inverse tangent of both sides, we get:

x = arctan(1.333...) ≈ 53.13 degrees

e. Using the reciprocal identity for cosine: cos(x) = 1/sec(x), we have:

cos(x) = 1/sec(60) = 1/2

Taking the inverse cosine of both sides, we get:

x = arccos(1/2) = 60 degrees