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Chapter 5: Problem 20

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. $$(8,15)$$

Short Answer

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

Step by step solution

01

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 胃 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.
02

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 \( c = 17 \).
03

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 胃 = 'opposite'/'hypotenuse' = 15/17, \nCos 胃 = 'adjacent'/'hypotenuse' = 8/17, \nTan 胃 = 'opposite'/'adjacent' = 15/8, \nCsc 胃 (cosecant) = 1/(Sin 胃) = 17/15, \nSec 胃 (secant) = 1/(Cos 胃) = 17/8, \nCot 胃 (cotangent) = 1/(Tan 胃) = 8/15.

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