Trigonometry Examples

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

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.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Multiply $- 1$ by $1$ .

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

Step 1.6

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

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Step 1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.2

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

Multiply $- 1$ 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

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$\frac{0}{2}$

Step 2.3

Divide $0$ by $2$ .

$0$