Problem 5 - Precalculus Essentials

Question

In Exercises \(1-8,\) a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta\). $$(3,-3)$$

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions for the point (3,-3) are: \(\sin(\theta) = -1, \cos(\theta) = 1, \tan(\theta) = -1, \csc(\theta) = -1, \sec(\theta) = 1, \cot(\theta) = -1\)

Step 1: Calculate the distance from the origin

Calculate r which is the distance from the origin to the point (3,-3) using Pythagoras' theorem. It can be calculated like so: r= \(\sqrt{(3)^2 + (-3)^2}\).

Step 2: Find the exact values of sin, cos, and tan

Now, compute the trigonometric functions sin, cos, tan using the following formulas: \(\sin(\theta)= -3 / r, \cos(\theta)= 3 / r, \tan(\theta)= \sin(\theta)/\cos(\theta)\).

Step 3: Find the exact values of csc, sec, cot

Compute the trigonometric functions csc, sec, cot which are the reciprocals of sin, cos, tan respectively, using the following formulas \(\csc(\theta)= 1 / \sin(\theta), \sec(\theta)= 1 / \cos(\theta), \cot(\theta)= 1 / \tan(\theta)\).