Trigonometry Examples

$\sin (\frac{7 π}{4}) \cos (2 π) - \cos (\frac{7 π}{4}) \sin (\frac{3 π}{2})$

Step 1

Simplify each term.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- \sin (\frac{π}{4}) \cos (2 π) - \cos (\frac{7 π}{4}) \sin (\frac{3 π}{2})$

Step 1.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} \cos (2 π) - \cos (\frac{7 π}{4}) \sin (\frac{3 π}{2})$

Step 1.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- \frac{\sqrt{2}}{2} \cos (0) - \cos (\frac{7 π}{4}) \sin (\frac{3 π}{2})$

Step 1.4

The exact value of $\cos (0)$ is $1$ .

$- \frac{\sqrt{2}}{2} \cdot 1 - \cos (\frac{7 π}{4}) \sin (\frac{3 π}{2})$

Step 1.5

Multiply $- 1$ by $1$ .

$- \frac{\sqrt{2}}{2} - \cos (\frac{7 π}{4}) \sin (\frac{3 π}{2})$

Step 1.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \frac{\sqrt{2}}{2} - \cos (\frac{π}{4}) \sin (\frac{3 π}{2})$

Step 1.7

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{3 π}{2})$

Step 1.8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} (- \sin (\frac{π}{2}))$

Step 1.9

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} (- 1 \cdot 1)$

Step 1.10

Multiply $- 1$ by $1$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot - 1$

Step 1.11

Multiply $- \frac{\sqrt{2}}{2} \cdot - 1$ .

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Step 1.11.1

Multiply $- 1$ by $- 1$ .

$- \frac{\sqrt{2}}{2} + 1 \frac{\sqrt{2}}{2}$

Step 1.11.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$- \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{- \sqrt{2} + \sqrt{2}}{2}$

Step 2.2

Add $- \sqrt{2}$ and $\sqrt{2}$ .

$\frac{0}{2}$

Step 2.3

Divide $0$ by $2$ .

$0$