Use the fundamental identities to find the value of the trigonometric function.
-$- 8 x - 5 y = 0 , x \geq 0 ; \quad$ Find $\cos \theta$
A) $\frac { 5 \sqrt { 89 } } { 89 }$
B) $- \frac { 5 } { 8 }$
C) $- \frac { 8 } { 5 }$
D) $\frac { \sqrt { 89 } } { 5 }$

Question

Quizplus · Accepted Answer

Given the equation $-8x - 5y = 0$, we can rearrange it to $y = -\frac{8}{5}x$. This equation represents a right triangle where $y$ is opposite to $\theta$, $x$ is adjacent to $\theta$, and the hypotenuse can be found using the Pythagorean theorem. The length of the hypotenuse is $\sqrt{x^2 + y^2} = \sqrt{x^2 + \left(-\frac{8}{5}x\right)^2} = \sqrt{x^2 + \frac{64}{25}x^2} = \sqrt{\frac{89}{25}x^2} = \frac{\sqrt{89}}{5}x$. Therefore, $\cos\theta = \frac{x}{\text{hypotenuse}} = \frac{x}{\frac{\sqrt{89}}{5}x} = \frac{5}{\sqrt{89}}$, which simplifies to $\frac{5\sqrt{89}}{89}$ when rationalizing the denominator.

Use the Fundamental Identities to Find the Value of the Trigonometric