QuestionFind the six trigonometric functions for the angle $-\frac{5 \pi}{4}$ without using a calculator.

Studdy Solution

STEP 1

Assumptions1. The given angle is $-\frac{5 \pi}{4}$ . We are looking for the exact values of the six trigonometric functions sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
3. The angle is in radians.

STEP 2

First, we need to find an equivalent positive angle for $-\frac{5 \pi}{4}$ because it's easier to work with positive angles. We can do this by adding $2\pi$ (a full circle) to the given angle.
$\theta = -\frac{5 \pi}{4} +2\pi$

STEP 3

Calculate the equivalent positive angle.
$\theta = -\frac{5 \pi}{} +2\pi = \frac{3 \pi}{}$

STEP 4

Now, we can find the exact values of the six trigonometric functions for the angle $\frac{3 \pi}{4}$ .The angle $\frac{3 \pi}{4}$ is in the second quadrant where sine is positive and cosine is negative.The values of sine, cosine, and tangent for the angles $\frac{\pi}{4}$ , $\frac{3 \pi}{4}$ , $\frac{ \pi}{4}$ , and $\frac{7 \pi}{4}$ are known values.For $\frac{\pi}{4}$ , we have $\sin{\frac{\pi}{4}} = \cos{\frac{\pi}{4}} = \frac{\sqrt{2}}{2}$ $\tan{\frac{\pi}{4}} =1$

STEP 5

For $\frac{3 \pi}{4}$ , we have $\sin{\frac{3 \pi}{4}} = \frac{\sqrt{2}}{2}$ $\cos{\frac{3 \pi}{4}} = -\frac{\sqrt{2}}{2}$ $\tan{\frac{3 \pi}{4}} = -1$

STEP 6

The reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.So, we can find the values of cosecant, secant, and cotangent for the angle $\frac{3 \pi}{4}$ by taking the reciprocals of the sine, cosine, and tangent values respectively.
$\csc{\frac{3 \pi}{4}} = \frac{1}{\sin{\frac{3 \pi}{4}}}$ $\sec{\frac{3 \pi}{4}} = \frac{1}{\cos{\frac{3 \pi}{4}}}$ $\cot{\frac{3 \pi}{4}} = \frac{1}{\tan{\frac{3 \pi}{4}}}$

STEP 7

Calculate the values of cosecant, secant, and cotangent for the angle $\frac{3 \pi}{4}$ .
$\csc{\frac{3 \pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$ $\sec{\frac{3 \pi}{4}} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}$ $\cot{\frac{3 \pi}{4}} = \frac{1}{-1} = -1$ The exact values of the six trigonometric functions for the angle $-\frac{5 \pi}{4}$ are $\sin{\left(-\frac{5 \pi}{4}\right)} = \frac{\sqrt{2}}{2}$ $\cos{\left(-\frac{5 \pi}{4}\right)} = -\frac{\sqrt{2}}{2}$ $\tan{\left(-\frac{5 \pi}{4}\right)} = -1$ $\csc{\left(-\frac{5 \pi}{4}\right)} = \sqrt{2}$ $\sec{\left(-\frac{5 \pi}{4}\right)} = -\sqrt{2}$ $\cot{\left(-\frac{5 \pi}{4}\right)} = -1$

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