Precalculus Examples

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

Step 1

Use the difference formula for cosine to simplify the expression. The formula states that $\cos (A - B) = \cos (A) \cos (B) + \sin (A) \sin (B)$ .

$\cos (\frac{3 π}{4}) \cdot \cos (\frac{7 π}{4}) + \sin (\frac{3 π}{4}) \cdot \sin (\frac{7 π}{4})$

Step 2

Remove parentheses.

$\cos (\frac{3 π}{4}) \cdot \cos (\frac{7 π}{4}) + \sin (\frac{3 π}{4}) \cdot \sin (\frac{7 π}{4})$

Step 3

Simplify each term.

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Step 3.1

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.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.5.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.5

Add $1$ and $1$ .

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

Step 3.5.6

Multiply $2$ by $2$ .

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

Step 3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{2^{\frac{2}{2}}}{4} + \sin (\frac{3 π}{4}) \cdot \sin (\frac{7 π}{4})$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$- \frac{2^{\frac{2}{2}}}{4} + \sin (\frac{3 π}{4}) \cdot \sin (\frac{7 π}{4})$

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Evaluate the exponent.

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

Step 3.7

Cancel the common factor of $2$ and $4$ .

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Step 3.7.1

Factor $2$ out of $2$ .

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

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $2$ out of $4$ .

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

Step 3.7.2.2

Cancel the common factor.

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

Step 3.7.2.3

Rewrite the expression.

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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.

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

Step 3.11

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

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

Step 3.12

Multiply $\frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2})$ .

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Step 3.12.1

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

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

Step 3.12.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.12.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.12.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.12.5

Add $1$ and $1$ .

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

Step 3.12.6

Multiply $2$ by $2$ .

$- \frac{1}{2} - \frac{{\sqrt{2}}^{2}}{4}$

Step 3.13

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.13.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$- \frac{1}{2} - \frac{{(2^{\frac{1}{2}})}^{2}}{4}$

Step 3.13.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{2} - \frac{2^{\frac{1}{2} \cdot 2}}{4}$

Step 3.13.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{1}{2} - \frac{2^{\frac{2}{2}}}{4}$

Step 3.13.4

Cancel the common factor of $2$ .

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Step 3.13.4.1

Cancel the common factor.

$- \frac{1}{2} - \frac{2^{\frac{2}{2}}}{4}$

Step 3.13.4.2

Rewrite the expression.

$- \frac{1}{2} - \frac{2^{1}}{4}$

Step 3.13.5

Evaluate the exponent.

$- \frac{1}{2} - \frac{2}{4}$

Step 3.14

Cancel the common factor of $2$ and $4$ .

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Step 3.14.1

Factor $2$ out of $2$ .

$- \frac{1}{2} - \frac{2 (1)}{4}$

Step 3.14.2

Cancel the common factors.

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Step 3.14.2.1

Factor $2$ out of $4$ .

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

Step 3.14.2.2

Cancel the common factor.

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

Step 3.14.2.3

Rewrite the expression.

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

Step 4

Combine fractions.

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Step 4.1

Combine the numerators over the common denominator.

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

Step 4.2

Simplify the expression.

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Step 4.2.1

Subtract $1$ from $- 1$ .

$\frac{- 2}{2}$

Step 4.2.2

Divide $- 2$ by $2$ .

$- 1$