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. (―12 , ―5)

Accepted Answer

Step 1: Understand the problem. We are given a point (-12, -5) 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. Step 2: Sketch the angle. Since the point (-12, -5) lies in the third quadrant (both x and y are negative), the terminal side of $$ 	heta  is in $$the third quadrant. The angle $$ 	heta  is $$measured from the positive x-axis to this terminal side, moving counterclockwise. Step 3: Calculate the radius (or hypotenuse) $$ r $$ using the distance formula: $$ r = \sqrt{x^2 + y^2} = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} . $$Step 4: Find the six trigonometric functions using the definitions based on the coordinates and radius: 
- $$ \sin 	heta = \frac{y}{r} = \frac{-5}{r} $$
- $$ \cos 	heta = \frac{x}{r} = \frac{-12}{r} $$
- $$ 	an 	heta = \frac{y}{x} = \frac{-5}{-12} $$
- $$ \csc 	heta = \frac{r}{y} = \frac{r}{-5} $$
- $$ \sec 	heta = \frac{r}{x} = \frac{r}{-12} $$
- $$ \cot 	heta = \frac{x}{y} = \frac{-12}{-5} $$ Step 5: Rationalize denominators if necessary and simplify each trigonometric function to its simplest form, keeping in mind the signs based on the quadrant where the angle lies.

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. (―12 , ―5)

Risposta video verificata per un problema simile:

Concetti chiave

Standard Position of an Angle

Finding the Reference Angle and Least Positive Angle

Six Trigonometric Functions from Coordinates