Question

Given sinθ = 63/65 and the angle terminates in quadrant 2, find cosθ:

a) 4/5
b) -4/5
c) -63/65
d) 63/65

Answer 1

Given sinθ = 63/65 in quadrant 2, the cosine of θ can be calculated using the Pythagorean identity, which results in cosθ = -16/65 or option (b) -4/5.

Step-by-step explanation:

The question involves finding the cosine of an angle given the sine and the angle's quadrant. Since the sine of the angle θ is given by sinθ = 63/65 and the angle terminates in quadrant 2, we know that the cosine in this quadrant must be negative because in the unit circle only sine is positive in the second quadrant. We can utilize the Pythagorean identity sin²θ + cos²θ = 1. Substituting the given sinθ value we get:

sin²θ + cos²θ = 1

(63/65)² + cos²θ = 1

3969/4225 + cos²θ = 1

cos²θ = 1 - 3969/4225

cos²θ = 256/4225

cosθ = ±√(256/4225)

cosθ = ±(16/65)

Because the angle is in the second quadrant where cosine is negative, the correct answer is cosθ = -16/65, which is option (b) -4/5.