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.

$tan^{2} (x)+1=\frac{289}{225}$

$tan^{2} (x)+\frac{225}{225} =\frac{289}{225}$

$tan^{2}(x)=\frac{64}{225}$

tan x=$\frac{8}{15}$, Tangent in the fourth  quadrant is negative  so  the answer is instead

$tan x= -\frac{8}{15}$

We can find cotangent by taking the recipocial of tan so

$cot=-\frac{15}{8}$

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

$cos =\frac{15}{17}$

We can find sin by doing quoteint identies.

$\frac{sin x}{cos x} =tan x$

$\frac{ sin x}{15/17} =-\frac{8}{15}$

$sin x= -\frac{8}{17}$

To find csc, take reciprocal of sine.

$csc=\frac{-17}{8}$