Question

sketch an angle θ in standard position such that θ has the least possible positive measure and the point (6, - 8) is on the terminal side of θ. then find the exact values of the six trigonometric functions for θ. choose the correct graph below. a. graph a description b. graph b description c. graph c description

Explanation:

Step1: Recall the definition of an angle in standard - position

An angle $\theta$ in standard position has its vertex at the origin $(0,0)$ and its initial side along the positive $x$ - axis. The point $(6,-8)$ is in the fourth quadrant since $x = 6>0$ and $y=-8 < 0$.

Step2: Determine the correct graph

The terminal side of the angle $\theta$ with the least positive measure and the point $(6, - 8)$ on its terminal side should be in the fourth quadrant.
For the six trigonometric functions:
First, find the value of $r$ using the formula $r=\sqrt{x^{2}+y^{2}}$, where $x = 6$ and $y=-8$.
$r=\sqrt{6^{2}+(-8)^{2}}=\sqrt{36 + 64}=\sqrt{100}=10$.

• Sine function: $\sin\theta=\frac{y}{r}=\frac{-8}{10}=-\frac{4}{5}$.
• Cosine function: $\cos\theta=\frac{x}{r}=\frac{6}{10}=\frac{3}{5}$.
• Tangent function: $\tan\theta=\frac{y}{x}=\frac{-8}{6}=-\frac{4}{3}$.
• Cosecant function: $\csc\theta=\frac{r}{y}=\frac{10}{-8}=-\frac{5}{4}$.
• Secant function: $\sec\theta=\frac{r}{x}=\frac{10}{6}=\frac{5}{3}$.
• Cotangent function: $\cot\theta=\frac{x}{y}=\frac{6}{-8}=-\frac{3}{4}$.

The angle in standard - position with the point $(6,-8)$ on its terminal side has its terminal side in the fourth quadrant. The correct graph is the one where the terminal side of the angle is in the fourth quadrant.

Answer:

The correct graph is the one with the terminal side of the angle in the fourth quadrant.
$\sin\theta=-\frac{4}{5}$, $\cos\theta=\frac{3}{5}$, $\tan\theta = -\frac{4}{3}$, $\csc\theta=-\frac{5}{4}$, $\sec\theta=\frac{5}{3}$, $\cot\theta=-\frac{3}{4}$