QuestionFind the six trigonometric functions for the angle $\frac{3 \pi}{4}$ . State "not defined" if applicable.

Studdy Solution

STEP 1

Assumptions1. The given angle is in radians and is $\frac{3 \pi}{4}$ . . We are to find the exact values of the six trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent.
3. If any of the trigonometric functions are not defined for the given angle, we will state "not defined."

STEP 2

We will start by finding the sine and cosine of the given angle. We know that for an angle $\theta$ in the unit circle, the sine is the y-coordinate and the cosine is the x-coordinate. For $\frac{ \pi}{4}$ , which is equivalent to $135^{\circ}$ , we can use the unit circle to find these values.
The sine and cosine of $\frac{ \pi}{4}$ are as follows $\sin\left(\frac{ \pi}{4}\right) = \frac{\sqrt{2}}{2}$ $\cos\left(\frac{ \pi}{4}\right) = -\frac{\sqrt{2}}{2}$

STEP 3

Next, we will find the tangent of the given angle. The tangent of an angle is the ratio of the sine to the cosine.
$\tan\left(\frac{3 \pi}{}\right) = \frac{\sin\left(\frac{3 \pi}{}\right)}{\cos\left(\frac{3 \pi}{}\right)}$

STEP 4

Substitute the values of sine and cosine from Step2 into the equation from Step3.
$\tan\left(\frac{3 \pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}$

STEP 5

Calculate the value of the tangent.
$\tan\left(\frac{3 \pi}{4}\right) = -1$

STEP 6

Next, we will find the values of the reciprocal trigonometric functions cosecant, secant, and cotangent. These are the reciprocals of sine, cosine, and tangent, respectively.
$\csc\left(\frac{3 \pi}{4}\right) = \frac{1}{\sin\left(\frac{3 \pi}{4}\right)}$ $\sec\left(\frac{3 \pi}{4}\right) = \frac{1}{\cos\left(\frac{3 \pi}{4}\right)}$ $\cot\left(\frac{3 \pi}{4}\right) = \frac{1}{\tan\left(\frac{3 \pi}{4}\right)}$

STEP 7

Substitute the values of sine, cosine, and tangent from Steps2 and5 into the equations from Step6.
$\csc\left(\frac{3 \pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}}$ $\sec\left(\frac{3 \pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}}$ $\cot\left(\frac{3 \pi}{4}\right) = \frac{1}{-1}$

STEP 8

Calculate the values of the reciprocal trigonometric functions.
$\csc\left(\frac{3 \pi}{4}\right) = \sqrt{2}$ $\sec\left(\frac{3 \pi}{4}\right) = -\sqrt{2}$ $\cot\left(\frac{3 \pi}{4}\right) = -1$ So, the exact values of the six trigonometric functions of $\frac{3 \pi}{4}$ are $\sin\left(\frac{3 \pi}{4}\right) = \frac{\sqrt{2}}{2}$ $\cos\left(\frac{3 \pi}{4}\right) = -\frac{\sqrt{2}}{2}$ $\tan\left(\frac{3 \pi}{4}\right) = -1$ $\csc\left(\frac{3 \pi}{4}\right) = \sqrt{2}$ $\sec\left(\frac{3 \pi}{4}\right) = -\sqrt{2}$ $\cot\left(\frac{3 \pi}{4}\right) = -1$

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