Question

Assume that θ is an acute angle in a right triangle satisfying the given conditions. Evaluate the 5 remaining trigonometric functions. Sketch a triangle in the correct quadrant and label.

sec θ= 17/15
sin θ<0

Answer 1

Answer: See step by step. Replace the x with theta

Step-by-step explanation: Since sin is less than zero, and secant is positive. This means the trig values are in Quadrant 4.

We know that secФ=17/15. We need to find sin,cos,tan,csc, and cot.

We can find tangent by using the identity

`tan^(2) (x)+ 1=sec^(2) (x)` where x is theta.

`tan^(2) (x)+1=(289)/(225)`

`tan^(2) (x)+(225)/(225) =(289)/(225)`

`tan^(2)(x)=(64)/(225)`

tan x=
`(8)/(15)`, Tangent in the fourth quadrant is negative so the answer is instead

`tan x= -(8)/(15)`

We can find cotangent by taking the recipocial of tan so

`cot=-(15)/(8)`

We can find cos by taking reciprocal of sec remeber that cosine is positive in 4th quadrant so the answer is

`cos =(15)/(17)`

We can find sin by doing quoteint identies.

`(sin x)/(cos x) =tan x`

`( sin x)/(15/17) =-(8)/(15)`

`sin x= -(8)/(17)`

To find csc, take reciprocal of sine.

`csc=(-17)/(8)`