Trigonometry Examples

$(\sqrt{4} \cos (45)) (\sqrt{4} \cos (135))$

Step 1

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2}} \cos (45) (\sqrt{4} \cos (135))$

Step 2

Pull terms out from under the radical, assuming positive real numbers.

$2 \cos (45) (\sqrt{4} \cos (135))$

Step 3

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$2 \frac{\sqrt{2}}{2} (\sqrt{4} \cos (135))$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

$2 \frac{\sqrt{2}}{2} (\sqrt{4} \cos (135))$

Step 4.2

Rewrite the expression.

$\sqrt{2} (\sqrt{4} \cos (135))$

Step 5

Rewrite $4$ as $2^{2}$ .

$\sqrt{2} (\sqrt{2^{2}} \cos (135))$

Step 6

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{2} (2 \cos (135))$

Step 7

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.

$\sqrt{2} (2 (- \cos (45)))$

Step 8

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

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

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

$\sqrt{2} (- \sqrt{2})$

Step 10

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

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Step 10.1

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

$- ({\sqrt{2}}^{1} \sqrt{2})$

Step 10.2

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

$- ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 10.3

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

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

Step 10.4

Add $1$ and $1$ .

$- {\sqrt{2}}^{2}$

Step 11

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

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

Cancel the common factor of $2$ .

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Step 11.4.1

Cancel the common factor.

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

Step 11.4.2

Rewrite the expression.

$- 2^{1}$

Step 11.5

Evaluate the exponent.

$- 1 \cdot 2$

Step 12

Multiply $- 1$ by $2$ .

$- 2$