Math question: reference angles and trigonometry

Question

Find the exact value of each of the remaining trigonometric functions of

θ.

tanθ= -2/7 cscθ>0

sinθ=

cosθ=

cscθ=

secθ=

cotθ=

Answer 1

Tangent is only negative in quadrant 2 and 4

Cosecant is the reciprocal of sine so both have the same sign, so sine is positive. Sine is not positive in quadrant 4, so our angle θ has to be in quadrant 2.

Since tanθ = -2/7 (opposite side over adjacent side), and θ is in quadrant 2

Use Pythagorean Theorem to find the hypoteneuse

c²=2²+7²

c²=4+49=53

c=√53

So,

sinθ = 2/√53 = (2√53)/53 rationalizing the denominator

cosθ = -7/√53 = (-7√53)/53 rationalizing the denominator

tanθ = 2/-7√

cscθ = √53/2√

secθ = √53/-7

cotθ = -7/2