Problem 7 - 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 \(\theta\). $$(-2,-5)$$

Step-by-Step Solution

Answer

sin(θ) = -5/ \(\sqrt{29}\), cos(θ) = -2/ \(\sqrt{29}\), tan(θ) = 5/2, csc(θ) = \(\sqrt{29}\)/-5, sec(θ) = \(\sqrt{29}\)/ -2, cot(θ) = 2/5.

Step 1: Identify the coordinates

The given point, (-2, -5), will represent the coordinates on the terminal side of the angle θ. Here, the x-coordinate is -2 and the y-coordinate is -5.

Step 2: Calculate the hypotenuse r

From the coordinates, calculate the hypotenuse (r) deriving from Pythagoras theorem as follows: \( r = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \). The hypotenuse or radius r is \(\sqrt{29}\).

Step 3: Express the trigonometric functions

The six trig functions are expressed as: sin(θ) = y/r, cos(θ) = x/r, tan(θ) = y/x, csc(θ) = r/y, sec(θ) = r/x, cot(θ) = x/y. Replace x with -2, y with -5 and r with \(\sqrt{29}\).

Step 4: Substitute the values into the formulas

Substitute -2, -5 and \(\sqrt{29}\) for x, y and r respectively in all the six trigonometric functions. Simplify where necessary to get the exact answers.