Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Feb 08, 2024

Compute $\sin(225^\circ)$ and $\cos(225^\circ)$ using the reference angle. The reference angle is $45^\circ$ , and the angle is in quadrant 2. $\sin(225^\circ) = -\sqrt{2}/2$ $\cos(225^\circ) = -\sqrt{2}/2$

STEP 1

Assumptions
1. We are working in degrees, not radians.
2. The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis.
3. The sine and cosine functions have specific signs depending on the quadrant in which the angle lies.
4. The angle $225^{\circ}$ is more than $180^{\circ}$ but less than $270^{\circ}$ , placing it in the third quadrant.
5. In the third quadrant, both sine and cosine are negative.
6. The reference angle for an angle in the third quadrant is $angle - 180^{\circ}$ .

STEP 2

Find the reference angle for $225^{\circ}$ . Since $225^{\circ}$ is in the third quadrant, we subtract $180^{\circ}$ to find the reference angle.
$Reference\, angle = 225^{\circ} - 180^{\circ}$

STEP 3

Calculate the reference angle.
$Reference\, angle = 225^{\circ} - 180^{\circ} = 45^{\circ}$
The reference angle is $45^{\circ}$ .

STEP 4

Identify the quadrant in which $225^{\circ}$ lies. Since it is more than $180^{\circ}$ but less than $270^{\circ}$ , it lies in the third quadrant.
The angle $225^{\circ}$ is in the third quadrant.

STEP 5

Use the reference angle to find the sine and cosine of $225^{\circ}$ . The sine and cosine of $45^{\circ}$ are known to be $\frac{\sqrt{2}}{2}$ .
$\sin(45^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2}$

STEP 6

Since $225^{\circ}$ is in the third quadrant, both sine and cosine are negative. Apply this to the values obtained from the reference angle.
$\sin(225^{\circ}) = -\sin(45^{\circ}) = -\frac{\sqrt{2}}{2}$ $\cos(225^{\circ}) = -\cos(45^{\circ}) = -\frac{\sqrt{2}}{2}$

STEP 7

Write the final answers.
$\begin{array}{l} \sin \left(225^{\circ}\right) = -\frac{\sqrt{2}}{2} \\ \cos \left(225^{\circ}\right) = -\frac{\sqrt{2}}{2} \end{array}$
The reference angle is $45^{\circ}$ and the angle is in the third quadrant. $\sin \left(225^{\circ}\right) = -\frac{\sqrt{2}}{2} \\ \cos \left(225^{\circ}\right) = -\frac{\sqrt{2}}{2}$

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