Question

Find the exact value of the trigonometric functions at the indicated angle. \(\csc t, \sec t, \operatorname{and} \cot t\) for \(t=1 / \pi / 6\)

Short answer

The exact values for the given trigonometric functions at t = 17π/6 are: Cosecant: \( \csc{t} = 2 \) Secant: \( \sec{t} = -\frac{2\sqrt{3}}{3} \) Cotangent: \( \cot{t} = -\sqrt{3} \)

Step-by-step solution

Find the Reference Angle

First, determine the reference angle θ, which is the angle between t and the closest x-axis. To do this, we find the coterminal angle in the range of 0 to 2π by calculating the remainder when t is divided by 2π, and then we find the reference angle by finding the smallest angle between the coterminal angle and the x-axis. The given angle, t, is 17π/6. To find its coterminal angle in the range of 0 to 2π, we calculate: \( \frac{17\pi}{6} \mathrm{mod\,} 2\pi = \frac{5\pi}{6} \) Now, the reference angle θ can be calculated as: \( \theta = \frac{5\pi}{6} \)

Determine the Quadrant

The given angle t is equal to 17π/6, and its coterminal angle, 5π/6, lies in the second quadrant because it's between π/2 and π.

Determine the Signs of the Trigonometric Functions

In the second quadrant, sine is positive while cosine and tangent are negative. Therefore, cosecant (the reciprocal of sine) will be positive, secant (the reciprocal of cosine) will be negative, and cotangent (the reciprocal of tangent) will also be negative.

Find the Values of Sine and Cosine

To find the exact values of cosecant and secant, we first need to find sine and cosine at the reference angle, θ = 5π/6. We can do this using the unit circle: \( \sin{\frac{5\pi}{6}} = \frac{1}{2} \) \( \cos{\frac{5\pi}{6}} = -\frac{\sqrt{3}}{2} \)

Calculate the Values of Cosecant, Secant, and Cotangent

Now we can find the values of the trigonometric functions for t = 17π/6 using the reference angle 5π/6 and the signs we determined in step 3: Cosecant (csc): Since the sine at the reference angle is 1/2 and cosecant is the reciprocal of sine, we have: \( \csc{\frac{17\pi}{6}} = \csc{\frac{5\pi}{6}} = \frac{1}{\sin{\frac{5\pi}{6}}} = \frac{1}{\frac{1}{2}} = 2 \) Secant (sec): Since the cosine at the reference angle is -√3/2 and secant is the reciprocal of cosine, we have: \( \sec{\frac{17\pi}{6}} = \sec{\frac{5\pi}{6}} = \frac{1}{\cos{\frac{5\pi}{6}}} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \) Cotangent (cot): Since cotangent is the reciprocal of tangent and tangent = sine/cosine, we have: \( \cot{\frac{17\pi}{6}} = \cot{\frac{5\pi}{6}} = \frac{1}{\tan{\frac{5\pi}{6}}} = \frac{\cos{\frac{5\pi}{6}}}{\sin{\frac{5\pi}{6}}} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3} \) So, the exact values for the given trigonometric functions at t = 17π/6 are: Cosecant: \( \csc{t} = 2 \) Secant: \( \sec{t} = -\frac{2\sqrt{3}}{3} \) Cotangent: \( \cot{t} = -\sqrt{3} \)