Question

The point (−3, 1) is on the terminal side of angle θ, in standard position. What are the values of sine, cosine, and tangent of θ? Make sure to show all work.
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Answer 1

see the attached figure to better understand the problem

Let

A(-3,1) B(0,0) C(-3,0)

we know that

the point A belong to the II quadrant

so

sin θ is positive

cos θ is negative

tan θ is negative

θ + α= 180 --------> by supplementary angles

in the right triangle ABC

sin α=opposite side angle α/hypotenuse

opposite side angle α=AC------> 1 units

hypotenuse----> AB------> applying the Pythagorean Theorem

AB²=AC²+BC²------> AB²=1²+3²-----> AB=√10 units

sin α=1/√10

cos α=adjacent side angle α/hypotenuse

adjacent side angle α=BC------> 3 units

hypotenuse----> AB------> √10 units

cos α=3/√10

tan α=opposite side angle α/adjacent side angle α

opposite side angle α=AC------> 1 units

adjacent side angle α=BC------> 3 units

tan α=1/3

Find angle α

α=arc tan (1/3)------>α=18.43°

Find angle θ

θ=180°-α------> θ=180-18.43-------> θ=161.57°

Hence

the answer is

θ=161.57°

α=18.43°

sin θ=+1/√10

cos θ=-(3/√10)

tan θ=-1/3