Question

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―8 , 15)

Accepted Answer

Step 1: Understand the problem. You are given a point (-8, 15) on the terminal side of an angle $$ 	heta  in $$standard position. The goal is to sketch this angle and find the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent for $$ 	heta . $$Step 2: Calculate the radius $$ r $$, which is the distance from the origin to the point $$(-8, 15). $$Use the distance formula:

$$ r = \sqrt{x^2 + y^2} = \sqrt{(-8)^2 + 15^2} $$ Step 3: Find the six trigonometric functions using the definitions based on the coordinates and radius:

- $$ \sin 	heta = \frac{y}{r} $$
- $$ \cos 	heta = \frac{x}{r} $$
- $$ 	an 	heta = \frac{y}{x} $$
- $$ \csc 	heta = \frac{r}{y} $$
- $$ \sec 	heta = \frac{r}{x} $$
- $$ \cot 	heta = \frac{x}{y} $$ Step 4: Determine the quadrant of the angle $$ 	heta . $$Since $$ x = -8  ($$negative) and $$ y = 15  ($$positive), the point lies in the second quadrant. This helps understand the signs of the trigonometric functions. Step 5: Rationalize denominators if any of the trigonometric function values have radicals in the denominator, and express all six functions in simplest form with correct signs based on the quadrant.

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―8 , 15)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Standard Position of an Angle

Finding the Reference Angle and Least Positive Angle

Six Trigonometric Functions from a Point on the Terminal Side