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Chapter 4: Problem 27

Evaluate (if possible) the six trigonometric functions at the real number. $$t=-5 \pi / 3$$

Short Answer

Expert verified
The values are: \(\sin(-5\pi/3) = \sqrt{3}/2, \cos(-5 \pi / 3) = 1/2, \tan(-5\pi/3) = \sqrt{3}, \csc(-5\pi/3) = 2\sqrt{3}/3, \sec(-5\pi/3) = 2\), and \(\cot(-5\pi/3) = 1/\sqrt{3}\).

Step by step solution

01

Positive Angle Conversion

Because trigonometric functions are periodic, positive and negative values can be used interchangeably. In order to make the value positive, add \(2\pi\) until the value is in the range \([0, 2\pi]\). So, \(t = -5\pi/3\) changes to \(t = \pi/3\) because \(-5\pi/3 + 2\pi = \pi/3\).
02

Evaluating Sine and Cosine

The sine and cosine values for \(t = \pi/3\) are widely known in trigonometry, and found in the first quadrant. So, \(\sin(\pi/3) = \sqrt{3}/2\) and \(\cos(\pi / 3) = 1/2\).
03

Evaluating Tangent

The tangent function is the ratio of sine to cosine. Therefore, \(\tan(\pi/3) = \sin(\pi/3) / \cos(\pi/3) = \sqrt{3}/2 / 1/2 = \sqrt{3}\).
04

Evaluating Cosecant, Secant, and Cotangent

The cosecant, secant and cotangent functions are reciprocal to sine, cosine and tangent respectively. Therefore, \(\csc(\pi/3) = 1/\sin(\pi/3) = 2/\sqrt{3} = 2\sqrt{3}/3, \sec(\pi/3) = 1/\cos(\pi/3) = 2\) and \(\cot(\pi/3) = 1/\tan(\pi/3) = 1/\sqrt{3}\).

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