Trigonometry Examples

$\sin^{2} (150 °) - \cos^{2} (150 °)$

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.

$\sin^{2} (30) - \cos^{2} (150 °)$

Step 1.2

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

${(\frac{1}{2})}^{2} - \cos^{2} (150 °)$

Step 1.3

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{2}}{2^{2}} - \cos^{2} (150 °)$

Step 1.4

One to any power is one.

$\frac{1}{2^{2}} - \cos^{2} (150 °)$

Step 1.5

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

$\frac{1}{4} - \cos^{2} (150 °)$

Step 1.6

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.

$\frac{1}{4} - {(- \cos (30))}^{2}$

Step 1.7

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{4} - {(- \frac{\sqrt{3}}{2})}^{2}$

Step 1.8

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

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Step 1.8.1

Apply the product rule to $- \frac{\sqrt{3}}{2}$ .

$\frac{1}{4} - ({(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2})$

Step 1.8.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{1}{4} - ({(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}})$

Step 1.9

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.9.1

Move ${(- 1)}^{2}$ .

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

Step 1.9.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.9.2.1

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

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

Step 1.9.2.2

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

$\frac{1}{4} + {(- 1)}^{2 + 1} \frac{{\sqrt{3}}^{2}}{2^{2}}$

Step 1.9.3

Add $2$ and $1$ .

$\frac{1}{4} + {(- 1)}^{3} \frac{{\sqrt{3}}^{2}}{2^{2}}$

Step 1.10

Raise $- 1$ to the power of $3$ .

$\frac{1}{4} - \frac{{\sqrt{3}}^{2}}{2^{2}}$

Step 1.11

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{4} - \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 1.11.2

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

$\frac{1}{4} - \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 1.11.3

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

$\frac{1}{4} - \frac{3^{\frac{2}{2}}}{2^{2}}$

Step 1.11.4

Cancel the common factor of $2$ .

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Step 1.11.4.1

Cancel the common factor.

$\frac{1}{4} - \frac{3^{\frac{2}{2}}}{2^{2}}$

Step 1.11.4.2

Rewrite the expression.

$\frac{1}{4} - \frac{3^{1}}{2^{2}}$

Step 1.11.5

Evaluate the exponent.

$\frac{1}{4} - \frac{3}{2^{2}}$

Step 1.12

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

$\frac{1}{4} - \frac{3}{4}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{1 - 3}{4}$

Step 2.2

Subtract $3$ from $1$ .

$\frac{- 2}{4}$

Step 2.3

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

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Step 2.3.1

Factor $2$ out of $- 2$ .

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

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $4$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

$\frac{- 1}{2}$

Step 2.4

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$