Solved on Jan 18, 2024

Evaluate the trigonometric functions of $\frac{7\pi}{4}$ .

STEP 1

Assumptions
1. We are working with trigonometric functions of an angle measured in radians.
2. We will use the unit circle and the properties of trigonometric functions to find the values.
3. The angle $\frac{7\pi}{4}$ is an angle that corresponds to a commonly used angle in the unit circle, which is $45^\circ$ or $\frac{\pi}{4}$ radians, but it is in the fourth quadrant.

STEP 2

To find $\sin\left(\frac{7\pi}{4}\right)$ , we need to determine the y-coordinate of the point on the unit circle at an angle of $\frac{7\pi}{4}$ radians from the positive x-axis.

STEP 3

Since $\frac{7\pi}{4}$ is an angle that is $2\pi - \frac{\pi}{4}$ , it is coterminal with $-\frac{\pi}{4}$ , which means it has the same sine value as $\frac{\pi}{4}$ but negative because it is in the fourth quadrant where sine is negative.

STEP 4

The sine of $\frac{\pi}{4}$ is $\frac{\sqrt{2}}{2}$ , so the sine of $\frac{7\pi}{4}$ is $-\frac{\sqrt{2}}{2}$ .
$\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

STEP 5

To find $\csc\left(\frac{7\pi}{4}\right)$ , we take the reciprocal of the sine value.

STEP 6

The cosecant is the reciprocal of sine, so $\csc\left(\frac{7\pi}{4}\right) = \frac{1}{\sin\left(\frac{7\pi}{4}\right)}$ .

STEP 7

Compute the cosecant value.
$\csc\left(\frac{7\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}$

STEP 8

To find $\cos\left(\frac{7\pi}{4}\right)$ , we determine the x-coordinate of the point on the unit circle at an angle of $\frac{7\pi}{4}$ radians from the positive x-axis.

STEP 9

The cosine of $\frac{\pi}{4}$ is $\frac{\sqrt{2}}{2}$ , and since cosine is positive in the fourth quadrant, the cosine of $\frac{7\pi}{4}$ is also $\frac{\sqrt{2}}{2}$ .
$\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}$

STEP 10

To find $\sec\left(\frac{7\pi}{4}\right)$ , we take the reciprocal of the cosine value.

STEP 11

The secant is the reciprocal of cosine, so $\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\cos\left(\frac{7\pi}{4}\right)}$ .

STEP 12

Compute the secant value.
$\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$

STEP 13

To find $\tan\left(\frac{7\pi}{4}\right)$ , we use the fact that tangent is the ratio of sine to cosine.

STEP 14

Compute the tangent value using the sine and cosine values we have found.
$\tan\left(\frac{7\pi}{4}\right) = \frac{\sin\left(\frac{7\pi}{4}\right)}{\cos\left(\frac{7\pi}{4}\right)}$

STEP 15

Substitute the sine and cosine values into the equation.
$\tan\left(\frac{7\pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$

STEP 16

Calculate the tangent value.
$\tan\left(\frac{7\pi}{4}\right) = -1$

STEP 17

To find $\cot\left(\frac{7\pi}{4}\right)$ , we take the reciprocal of the tangent value or use the ratio of cosine to sine.

STEP 18

Compute the cotangent value using the cosine and sine values we have found.
$\cot\left(\frac{7\pi}{4}\right) = \frac{\cos\left(\frac{7\pi}{4}\right)}{\sin\left(\frac{7\pi}{4}\right)}$

STEP 19

Substitute the sine and cosine values into the equation.
$\cot\left(\frac{7\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}$

STEP 20

Calculate the cotangent value.
$\cot\left(\frac{7\pi}{4}\right) = -1$
The solutions are: $\sin \left(\frac{7 \pi}{4}\right) = -\frac{\sqrt{2}}{2}, \quad \csc \left(\frac{7 \pi}{4}\right) = -\sqrt{2},$ $\cos \left(\frac{7 \pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \sec \left(\frac{7 \pi}{4}\right) = \sqrt{2},$ $\tan \left(\frac{7 \pi}{4}\right) = -1, \quad \cot \left(\frac{7 \pi}{4}\right) = -1.$

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Evaluate the trigonometric functions of 7π4\frac{7\pi}{4}47π​.

STEP 1

STEP 2

To find sin⁡(7π4)\sin\left(\frac{7\pi}{4}\right)sin(47π​), we need to determine the y-coordinate of the point on the unit circle at an angle of 7π4\frac{7\pi}{4}47π​ radians from the positive x-axis.

STEP 3

Since 7π4\frac{7\pi}{4}47π​ is an angle that is 2π−π42\pi - \frac{\pi}{4}2π−4π​, it is coterminal with −π4-\frac{\pi}{4}−4π​, which means it has the same sine value as π4\frac{\pi}{4}4π​ but negative because it is in the fourth quadrant where sine is negative.

STEP 4

The sine of π4\frac{\pi}{4}4π​ is 22\frac{\sqrt{2}}{2}22​​, so the sine of 7π4\frac{7\pi}{4}47π​ is −22-\frac{\sqrt{2}}{2}−22​​.sin⁡(7π4)=−22\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}sin(47π​)=−22​​

STEP 5

To find csc⁡(7π4)\csc\left(\frac{7\pi}{4}\right)csc(47π​), we take the reciprocal of the sine value.

STEP 6

The cosecant is the reciprocal of sine, so csc⁡(7π4)=1sin⁡(7π4)\csc\left(\frac{7\pi}{4}\right) = \frac{1}{\sin\left(\frac{7\pi}{4}\right)}csc(47π​)=sin(47π​)1​.

STEP 7

Compute the cosecant value.csc⁡(7π4)=1−22=−2\csc\left(\frac{7\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}csc(47π​)=−22​​1​=−2​

STEP 8

To find cos⁡(7π4)\cos\left(\frac{7\pi}{4}\right)cos(47π​), we determine the x-coordinate of the point on the unit circle at an angle of 7π4\frac{7\pi}{4}47π​ radians from the positive x-axis.

STEP 9

The cosine of π4\frac{\pi}{4}4π​ is 22\frac{\sqrt{2}}{2}22​​, and since cosine is positive in the fourth quadrant, the cosine of 7π4\frac{7\pi}{4}47π​ is also 22\frac{\sqrt{2}}{2}22​​.cos⁡(7π4)=22\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}cos(47π​)=22​​

STEP 10

To find sec⁡(7π4)\sec\left(\frac{7\pi}{4}\right)sec(47π​), we take the reciprocal of the cosine value.

STEP 11

The secant is the reciprocal of cosine, so sec⁡(7π4)=1cos⁡(7π4)\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\cos\left(\frac{7\pi}{4}\right)}sec(47π​)=cos(47π​)1​.

STEP 12

Compute the secant value.sec⁡(7π4)=122=2\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}sec(47π​)=22​​1​=2​

STEP 13

To find tan⁡(7π4)\tan\left(\frac{7\pi}{4}\right)tan(47π​), we use the fact that tangent is the ratio of sine to cosine.

STEP 14

Compute the tangent value using the sine and cosine values we have found.tan⁡(7π4)=sin⁡(7π4)cos⁡(7π4)\tan\left(\frac{7\pi}{4}\right) = \frac{\sin\left(\frac{7\pi}{4}\right)}{\cos\left(\frac{7\pi}{4}\right)}tan(47π​)=cos(47π​)sin(47π​)​

STEP 15

Substitute the sine and cosine values into the equation.tan⁡(7π4)=−2222\tan\left(\frac{7\pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}tan(47π​)=22​​−22​​​

STEP 16

Calculate the tangent value.tan⁡(7π4)=−1\tan\left(\frac{7\pi}{4}\right) = -1tan(47π​)=−1

STEP 17

To find cot⁡(7π4)\cot\left(\frac{7\pi}{4}\right)cot(47π​), we take the reciprocal of the tangent value or use the ratio of cosine to sine.

STEP 18

Compute the cotangent value using the cosine and sine values we have found.cot⁡(7π4)=cos⁡(7π4)sin⁡(7π4)\cot\left(\frac{7\pi}{4}\right) = \frac{\cos\left(\frac{7\pi}{4}\right)}{\sin\left(\frac{7\pi}{4}\right)}cot(47π​)=sin(47π​)cos(47π​)​

STEP 19

Substitute the sine and cosine values into the equation.cot⁡(7π4)=22−22\cot\left(\frac{7\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}cot(47π​)=−22​​22​​​

STEP 20

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Evaluate the trigonometric functions of $\frac{7\pi}{4}$ .

To find $\sin\left(\frac{7\pi}{4}\right)$ , we need to determine the y-coordinate of the point on the unit circle at an angle of $\frac{7\pi}{4}$ radians from the positive x-axis.

Since $\frac{7\pi}{4}$ is an angle that is $2\pi - \frac{\pi}{4}$ , it is coterminal with $-\frac{\pi}{4}$ , which means it has the same sine value as $\frac{\pi}{4}$ but negative because it is in the fourth quadrant where sine is negative.

The sine of $\frac{\pi}{4}$ is $\frac{\sqrt{2}}{2}$ , so the sine of $\frac{7\pi}{4}$ is $-\frac{\sqrt{2}}{2}$ .
$\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

To find $\csc\left(\frac{7\pi}{4}\right)$ , we take the reciprocal of the sine value.

The cosecant is the reciprocal of sine, so $\csc\left(\frac{7\pi}{4}\right) = \frac{1}{\sin\left(\frac{7\pi}{4}\right)}$ .

Compute the cosecant value.
$\csc\left(\frac{7\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}$

To find $\cos\left(\frac{7\pi}{4}\right)$ , we determine the x-coordinate of the point on the unit circle at an angle of $\frac{7\pi}{4}$ radians from the positive x-axis.

The cosine of $\frac{\pi}{4}$ is $\frac{\sqrt{2}}{2}$ , and since cosine is positive in the fourth quadrant, the cosine of $\frac{7\pi}{4}$ is also $\frac{\sqrt{2}}{2}$ .
$\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}$

To find $\sec\left(\frac{7\pi}{4}\right)$ , we take the reciprocal of the cosine value.

The secant is the reciprocal of cosine, so $\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\cos\left(\frac{7\pi}{4}\right)}$ .

Compute the secant value.
$\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$

To find $\tan\left(\frac{7\pi}{4}\right)$ , we use the fact that tangent is the ratio of sine to cosine.

Compute the tangent value using the sine and cosine values we have found.
$\tan\left(\frac{7\pi}{4}\right) = \frac{\sin\left(\frac{7\pi}{4}\right)}{\cos\left(\frac{7\pi}{4}\right)}$

Substitute the sine and cosine values into the equation.
$\tan\left(\frac{7\pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$

Calculate the tangent value.
$\tan\left(\frac{7\pi}{4}\right) = -1$

To find $\cot\left(\frac{7\pi}{4}\right)$ , we take the reciprocal of the tangent value or use the ratio of cosine to sine.

Compute the cotangent value using the cosine and sine values we have found.
$\cot\left(\frac{7\pi}{4}\right) = \frac{\cos\left(\frac{7\pi}{4}\right)}{\sin\left(\frac{7\pi}{4}\right)}$

Substitute the sine and cosine values into the equation.
$\cot\left(\frac{7\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}$