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Chapter 4: Problem 24

Evaluate (if possible) the six trigonometric functions at the real number. $$t=5 \pi / 6$$

Short Answer

Expert verified
The six trigonometric functions at \( t = 5\pi/6 \) are: \(\sin(5\pi/6) = 1/2, \cos(5\pi/6) = -\sqrt{3}/2, \tan(5\pi/6) = -\sqrt{3}/3, \csc(5\pi/6) = 2, \sec(5\pi/6) = -2/\sqrt{3}, \cot(5\pi/6) = -\sqrt{3}\)

Step by step solution

01

Position the given angle on the unit circle

The given angle \( t \) is \( 5\pi / 6 \), which is equivalent to \(150^\circ\). This angle is positioned in the second quadrant on the unit circle.
02

Analyze the values of sine, cosine and tangent

In the second quadrant, sine being the \(y\)-coordinate on the unit circle is positive, cosine being \(x\)-coordinate is negative and tangent being the ratio of \(y/x\) is negative. Also, they can also be described as follows: \(\sin(5\pi/6) = \sin(\pi/6) = 1/2, \cos(5\pi/6) = -\cos(\pi/6) = -\sqrt{3}/2, \tan(5\pi/6) = -\tan(\pi/6) = -\sqrt{3}/3\)
03

Compute the values of cosecant, secant, and cotangent

These are the reciprocal of sine, cosine and tangent and hence negative in case their corresponding primary trigonometric function is negative. So, \(\csc(5\pi/6) = \csc(\pi/6) = 2, \sec(5\pi/6) = -\sec(\pi/6) = -2/\sqrt{3}, \cot(5\pi/6) = -\cot(\pi/6) = -\sqrt{3}\)

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