Chapter 4.2, Problem 9E

To determine

Find the exact values of 6 trigonometric functions of the angle $\text{θ}$

Answer to Problem 9E

  $\text{sinθ}=\frac{\text{8}}{\text{17}};\text{cscθ}=\frac{17}{8}\text{;}\text{cosθ}=-\frac{\text{15}}{\text{17}};\text{secθ}=-\frac{\text{17}}{\text{15}};\text{tanθ}=-\frac{8}{15};\text{cotθ}=-\frac{15}{8}$

Explanation of Solution

Given: $x=-\frac{\text{15}}{\text{17}}$and $y=\frac{\text{8}}{\text{17}}$

Concept Used:

For each coordinates, begin by finding the corresponding point (x,y) on the unit circle. Then use the definition of trigonometric Functions.

For unit circle, the radius = 1

We know that in Trigonometry: $\text{sinθ}\text{=}y;\text{cosθ}\text{=}x;\text{tanθ}\text{=}\frac{\text{sinθ}}{\text{cosθ}}=\frac{\text{y}}{\text{x}}$

  $\text{cscθ}\text{=}\frac{\text{1}}{\text{sinθ}}=\frac{\text{1}}{\text{y}};\text{secθ}\text{=}\frac{\text{1}}{\text{cosθ}}=\frac{\text{1}}{\text{x}};\text{cotθ}\text{=}\frac{\text{1}}{\text{tanθ}}=\frac{\text{x}}{\text{y}}$

Calculation:

$\text{sinθ}\text{=}y=\frac{\text{8}}{\text{17}}$ ;   $\text{cscθ}\text{=}\frac{\text{1}}{\text{sinθ}}=\frac{17}{8}$   $\text{cosθ}\text{=}x=-\frac{\text{15}}{\text{17}}$;   $\text{secθ}\text{=}\frac{\text{1}}{\text{cosθ}}=-\frac{\text{17}}{\text{15}}$   $\text{tanθ}\text{=}\frac{\text{sinθ}}{\text{cosθ}}=-\frac{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$17$}\right.}{\raisebox{1ex}{$\text{15}$}\!\left/ \!\raisebox{-1ex}{$17$}\right.}=-\frac{8}{15}$ ;   $\text{cotθ}\text{=}\frac{\text{1}}{\text{tanθ}}=-\frac{15}{8}$

Thus, six Trigometric functions are: $\text{sinθ}=\frac{\text{8}}{\text{17}};\text{cscθ}=\frac{17}{8}\text{;}\text{cosθ}=-\frac{\text{15}}{\text{17}};\text{secθ}=-\frac{\text{17}}{\text{15}};\text{tanθ}=-\frac{8}{15};\text{cotθ}=-\frac{15}{8}$