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Chapter 4: Problem 2

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 \(\theta\). $$(-12,5)$$

Short Answer

Expert verified
The exact values of the six trigonometric functions for the given point (-12,5) are: \(sin\theta = \frac{5}{13}\), \(cos\theta = \frac{-12}{13}\), \(tan\theta = -\frac{5}{12}\), \(csc\theta = \frac{13}{5}\), \(sec\theta = -\frac{13}{12}\), \(cot\theta = -\frac{12}{5}\).

Step by step solution

01

Find the Radius r

Compute the radius \(r\) using the Pythagorean theorem. The radius is given by the formula \(r = \sqrt{x^{2} + y^{2}}\), where \(x = -12\) and \(y = 5\). Hence, \(r = \sqrt{(-12)^{2} + 5^{2}} = \sqrt{144 + 25} = \sqrt{169} = 13\).
02

Calculate Sine

The sine function (\(sin\theta\)) is given by the ratio of the y-coordinate to the radius. So, \(sin\theta = \frac{y}{r} = \frac{5}{13}\).
03

Calculate Cosine

The cosine function (\(cos\theta\)) is given by the ratio of the x-coordinate to the radius. So, \(cos\theta = \frac{x}{r} = \frac{-12}{13}\).
04

Calculate Tangent

The tangent function (\(tan\theta\)) is given by the ratio of sine to cosine. So, \(tan\theta = \frac{sin\theta}{cos\theta} = \frac{5/13}{-12/13} = -\frac{5}{12}\).
05

Calculate Cosecant

The cosecant function (\(csc\theta\)) is the reciprocal of sine. So, \(csc\theta = \frac{1}{sin\theta} = \frac{13}{5}\).
06

Calculate Secant

The secant function (\(sec\theta\)) is the reciprocal of cosine. So, \(sec\theta = \frac{1}{cos\theta} = -\frac{13}{12}\).
07

Calculate Cotangent

The cotangent function (\(cot\theta\)) is the reciprocal of tangent. So, \(cot\theta = \frac{1}{tan\theta} = -\frac{12}{5}\).

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