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Chapter 4: Problem 25

Evaluate (if possible) the six trigonometric functions at the real number. $$t=4 \pi / 3$$

Short Answer

Expert verified
So, the values of six trigonometric functions at \(t = 4\pi / 3\) are: \( \sin(4\pi / 3) = -\sqrt{3}/2\), \( \cos(4\pi / 3) = -1/2\), \( \tan(4\pi / 3) = \sqrt{3}\), \( \cot(4\pi / 3) = 1 / \sqrt{3}\), \( \sec(4\pi / 3) = -2\), \( \csc(4\pi / 3) = -2 / \sqrt{3}\).

Step by step solution

01

Find the Quadrant and corresponding angle

Firstly, we must determine at which quadrant and angle \(t = 4\pi / 3\) puts us at on the unit circle. Since \(4\pi / 3\) is a little more than \(3\pi / 2\) and less than \(2\pi\), this puts us in the 3rd quadrant for angle \(\pi/3\).
02

Evaluate Sine and Cosine functions

In this step, Cosine is negative in the 3rd quadrant and Sine is also negative. The angle \(\pi/3\) in the 3rd quadrant comes out to be \(-1/2\) for cosine and \(-\sqrt{3}/2\) for sine. Therefore, \( \cos(4\pi / 3) = -1/2\) and \( \sin(4\pi / 3) = -\sqrt{3}/2\).
03

Evaluate other Trigonometric functions

Once we have sine and cosine values, we can easily find the other trigonometric function values. \( \tan(4\pi / 3) = \sin(4\pi / 3) / \cos(4\pi / 3) = \sqrt{3}\). Cotangent is just the reciprocal of tangent so \( \cot(4\pi / 3) = 1/ \tan(4\pi / 3) = 1 / \sqrt{3}\). Lastly, secant and cosecant are the reciprocals of cosine and sine respectively, so \( \sec(4\pi / 3) = -2, \csc(4\pi / 3) = -2 / \sqrt{3}\).

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