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Chapter 4: Problem 14

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
The six trigonometric functions of the given angle are: \(\sin{\Theta} = 15/17\), \(\cos{\Theta} = 8/17\), \(\tan{\Theta} = 15/8\), \(\csc{\Theta} = 17/15\), \(\sec{\Theta} = 17/8\), and \(\cot{\Theta} = 8/15\).

Step by step solution

01

Visualization and Hypotenuse Calculation

First, visualize the point in the coordinate plane. The definition of a standard position angle is one whose initial side is on the positive x-axis. Consequently, the point (8,15) determines a right triangle with sides 8 (adjacent), 15 (opposite), and a hypotenuse which we calculate using the Pythagorean theorem: \(c = \sqrt{a^2 + b^2} = \sqrt{8^2 + 15^2} = \sqrt{289} = 17\).
02

Calculation of the Basic Trigonometric Functions

Now that the lengths of all sides are known, we can calculate the sine, cosine, and tangent of the angle. \(\sin{\Theta}=\)opposite/hypotenuse, \(\cos{\Theta}=\)adjacent/hypotenuse, and \(\tan{\Theta}=\)opposite/adjacent. This gives us \(\sin{\Theta} = 15/17\), \(\cos{\Theta} = 8/17\), \(\tan{\Theta} = 15/8\).
03

Calculation of the Reciprocal Trigonometric Functions

The remaining three trigonometric functions, cosecant, secant, and cotangent, are just the reciprocal of sine, cosine, and tangent respectively. This means \(\csc{\Theta} = 1/\sin{\Theta} = 17/15\), \(\sec{\Theta} = 1/\cos{\Theta} = 17/8\), and \(\cot{\Theta} = 1/\tan{\Theta} = 8/15\).

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