Find the exact value of each of the six trigonometric functions of θ​, if (4,−8) is a point on the terminal side of angle θ.

Answer 1

Use SOH CAH TOA, and their reciprocals.

Adjacent A is the x, A=4

Opposite O is the y, O=-8

Hypothenuse H is √(x²+y²) = √(16+64) = √(80) = 4√5

So

sin Θ = = -8/4√5 = -2/√5 = -2√5/5 (the last step, multiply by √5/√5 to "rationalize" the denominator)

cos Θ = 4/4√5 = 1/√5 = √5/5

tan Θ = -8/4 = -2

And their reciprocals

csc Θ = -√5/2

sec Θ = √5

cot Θ = -1/2