Math / Trigonometry

QuestionIf $\theta=\frac{7 \pi}{4}$ , then $\begin{array}{l} \sin (\theta)=\square \\ \cos (\theta)=\square \end{array}$
Give exact values. No decimals allowed! Question Help: $\square$ Video 1 $\square$ Video 2
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Studdy Solution

STEP 1

1. We are given the angle $\theta = \frac{7\pi}{4}$ .
2. We need to find the exact values of $\sin(\theta)$ and $\cos(\theta)$ .
3. The angle $\theta$ is in radians.

STEP 2

1. Determine the reference angle for $\theta = \frac{7\pi}{4}$ .
2. Identify the quadrant in which $\theta$ lies.
3. Use the reference angle to find $\sin(\theta)$ .
4. Use the reference angle to find $\cos(\theta)$ .

STEP 3

Calculate the reference angle for $\theta = \frac{7\pi}{4}$ .
The reference angle is calculated by finding the equivalent angle within the first circle, which is $2\pi - \theta$ .
$\text{Reference angle} = 2\pi - \frac{7\pi}{4} = \frac{8\pi}{4} - \frac{7\pi}{4} = \frac{\pi}{4}$

STEP 4

Identify the quadrant for $\theta = \frac{7\pi}{4}$ .
Since $\frac{7\pi}{4}$ is greater than $\frac{3\pi}{2}$ and less than $2\pi$ , $\theta$ is in the fourth quadrant.

STEP 5

Use the reference angle to find $\sin(\theta)$ .
In the fourth quadrant, the sine function is negative. Therefore:
$\sin\left(\frac{7\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

STEP 6

Use the reference angle to find $\cos(\theta)$ .
In the fourth quadrant, the cosine function is positive. Therefore:
$\cos\left(\frac{7\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
The exact values are:
$\sin(\theta) = -\frac{\sqrt{2}}{2}$
$\cos(\theta) = \frac{\sqrt{2}}{2}$

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QuestionIf $\theta=\frac{7 \pi}{4}$ , then $\begin{array}{l} \sin (\theta)=\square \\ \cos (\theta)=\square \end{array}$
Give exact values. No decimals allowed! Question Help: $\square$ Video 1 $\square$ Video 2
Check Answer

Studdy Solution

STEP 1

1. We are given the angle $\theta = \frac{7\pi}{4}$ .
2. We need to find the exact values of $\sin(\theta)$ and $\cos(\theta)$ .
3. The angle $\theta$ is in radians.

STEP 2

1. Determine the reference angle for $\theta = \frac{7\pi}{4}$ .
2. Identify the quadrant in which $\theta$ lies.
3. Use the reference angle to find $\sin(\theta)$ .
4. Use the reference angle to find $\cos(\theta)$ .

STEP 3

Calculate the reference angle for $\theta = \frac{7\pi}{4}$ .
The reference angle is calculated by finding the equivalent angle within the first circle, which is $2\pi - \theta$ .
$\text{Reference angle} = 2\pi - \frac{7\pi}{4} = \frac{8\pi}{4} - \frac{7\pi}{4} = \frac{\pi}{4}$

STEP 4

Identify the quadrant for $\theta = \frac{7\pi}{4}$ .
Since $\frac{7\pi}{4}$ is greater than $\frac{3\pi}{2}$ and less than $2\pi$ , $\theta$ is in the fourth quadrant.

STEP 5

Use the reference angle to find $\sin(\theta)$ .
In the fourth quadrant, the sine function is negative. Therefore:
$\sin\left(\frac{7\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

STEP 6

Use the reference angle to find $\cos(\theta)$ .
In the fourth quadrant, the cosine function is positive. Therefore:
$\cos\left(\frac{7\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
The exact values are:
$\sin(\theta) = -\frac{\sqrt{2}}{2}$
$\cos(\theta) = \frac{\sqrt{2}}{2}$

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QuestionIf θ=7π4\theta=\frac{7 \pi}{4}θ=47π​, then sin⁡(θ)=□cos⁡(θ)=□\begin{array}{l} \sin (\theta)=\square \\ \cos (\theta)=\square \end{array}sin(θ)=□cos(θ)=□​Give exact values. No decimals allowed! Question Help: □\square□ Video 1 □\square□ Video 2Check Answer

Studdy Solution

STEP 1

1. We are given the angle θ=7π4\theta = \frac{7\pi}{4}θ=47π​.2. We need to find the exact values of sin⁡(θ)\sin(\theta)sin(θ) and cos⁡(θ)\cos(\theta)cos(θ).3. The angle θ\thetaθ is in radians.

STEP 2

1. Determine the reference angle for θ=7π4\theta = \frac{7\pi}{4}θ=47π​.2. Identify the quadrant in which θ\thetaθ lies.3. Use the reference angle to find sin⁡(θ)\sin(\theta)sin(θ).4. Use the reference angle to find cos⁡(θ)\cos(\theta)cos(θ).

STEP 3

STEP 4

Identify the quadrant for θ=7π4\theta = \frac{7\pi}{4}θ=47π​.Since 7π4\frac{7\pi}{4}47π​ is greater than 3π2\frac{3\pi}{2}23π​ and less than 2π2\pi2π, θ\thetaθ is in the fourth quadrant.

STEP 5

Use the reference angle to find sin⁡(θ)\sin(\theta)sin(θ).In the fourth quadrant, the sine function is negative. Therefore:sin⁡(7π4)=−sin⁡(π4)=−22\sin\left(\frac{7\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}sin(47π​)=−sin(4π​)=−22​​

STEP 6

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QuestionIf $\theta=\frac{7 \pi}{4}$ , then $\begin{array}{l} \sin (\theta)=\square \\ \cos (\theta)=\square \end{array}$
Give exact values. No decimals allowed! Question Help: $\square$ Video 1 $\square$ Video 2
Check Answer

1. We are given the angle $\theta = \frac{7\pi}{4}$ .
2. We need to find the exact values of $\sin(\theta)$ and $\cos(\theta)$ .
3. The angle $\theta$ is in radians.

1. Determine the reference angle for $\theta = \frac{7\pi}{4}$ .
2. Identify the quadrant in which $\theta$ lies.
3. Use the reference angle to find $\sin(\theta)$ .
4. Use the reference angle to find $\cos(\theta)$ .

Identify the quadrant for $\theta = \frac{7\pi}{4}$ .
Since $\frac{7\pi}{4}$ is greater than $\frac{3\pi}{2}$ and less than $2\pi$ , $\theta$ is in the fourth quadrant.

Use the reference angle to find $\sin(\theta)$ .
In the fourth quadrant, the sine function is negative. Therefore:
$\sin\left(\frac{7\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$