Trigonometry Examples

$\cos^{2} (135) - 2 \cos (135) - 1$

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 cosine is negative in the second quadrant.

${(- \cos (45))}^{2} - 2 \cos (135) - 1$

Step 1.2

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

${(- \frac{\sqrt{2}}{2})}^{2} - 2 \cos (135) - 1$

Step 1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.3.1

Apply the product rule to $- \frac{\sqrt{2}}{2}$ .

${(- 1)}^{2} {(\frac{\sqrt{2}}{2})}^{2} - 2 \cos (135) - 1$

Step 1.3.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

${(- 1)}^{2} \frac{{\sqrt{2}}^{2}}{2^{2}} - 2 \cos (135) - 1$

Step 1.4

Raise $- 1$ to the power of $2$ .

$1 \frac{{\sqrt{2}}^{2}}{2^{2}} - 2 \cos (135) - 1$

Step 1.5

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

$\frac{{\sqrt{2}}^{2}}{2^{2}} - 2 \cos (135) - 1$

Step 1.6

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

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Step 1.6.1

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

$\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} - 2 \cos (135) - 1$

Step 1.6.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} - 2 \cos (135) - 1$

Step 1.6.3

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

$\frac{2^{\frac{2}{2}}}{2^{2}} - 2 \cos (135) - 1$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{2^{2}} - 2 \cos (135) - 1$

Step 1.6.4.2

Rewrite the expression.

$\frac{2^{1}}{2^{2}} - 2 \cos (135) - 1$

Step 1.6.5

Evaluate the exponent.

$\frac{2}{2^{2}} - 2 \cos (135) - 1$

Step 1.7

Raise $2$ to the power of $2$ .

$\frac{2}{4} - 2 \cos (135) - 1$

Step 1.8

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

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Step 1.8.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} - 2 \cos (135) - 1$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} - 2 \cos (135) - 1$

Step 1.8.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} - 2 \cos (135) - 1$

Step 1.8.2.3

Rewrite the expression.

$\frac{1}{2} - 2 \cos (135) - 1$

Step 1.9

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.

$\frac{1}{2} - 2 (- \cos (45)) - 1$

Step 1.10

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

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

Step 1.11

Cancel the common factor of $2$ .

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Step 1.11.1

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

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

Step 1.11.2

Factor $2$ out of $- 2$ .

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

Step 1.11.3

Cancel the common factor.

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

Step 1.11.4

Rewrite the expression.

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

Step 1.12

Multiply $- 1$ by $- 1$ .

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

Step 1.13

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

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

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

Subtract $2$ from $1$ .

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

Step 6

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.91421356 \dots$