Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(8,15)$$

Step-by-Step Solution

Answer

\nThe six trigonometric functions are: Sin θ = 15/17, Cos θ = 8/17, Tan θ = 15/8, Csc θ = 17/15, Sec θ = 17/8, Cot θ = 8/15.

Step 1: Write Down Given Co-ordinates and Pythagoras' theorem

The given point is (8,15). We can see it as coordinates of a right triangle where 8 is adjacent (cosine), and 15 is opposite (sine) to the angle θ we're looking at. Here, use \( c = \sqrt{a^2 + b^2} \) to find hypotenuse (radius) 'c', where 'a' and 'b' are sides of the triangle.

Step 2: Find the Radius

Substitute the values of 'a' and 'b' into the equation to get \( c = \sqrt{8^2 + 15^2} \). After doing the math, we find that \( c = 17 \).

Step 3: Calculate Trigonometric Values

All six trigonometric functions can be directly calculated from the given point (8,15) and radius 17, using their definitions:\n\nSin θ = 'opposite'/'hypotenuse' = 15/17, \nCos θ = 'adjacent'/'hypotenuse' = 8/17, \nTan θ = 'opposite'/'adjacent' = 15/8, \nCsc θ (cosecant) = 1/(Sin θ) = 17/15, \nSec θ (secant) = 1/(Cos θ) = 17/8, \nCot θ (cotangent) = 1/(Tan θ) = 8/15.