Trigonometry Examples

$\sin^{2} (60) - \cos^{2} (45) - \sin^{2} (30)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

${(\frac{\sqrt{3}}{2})}^{2} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.2

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

$\frac{{\sqrt{3}}^{2}}{2^{2}} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.3

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

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Step 1.3.1

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

$\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.3.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.3.3

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

$\frac{3^{\frac{2}{2}}}{2^{2}} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2^{2}} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.3.4.2

Rewrite the expression.

$\frac{3^{1}}{2^{2}} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.3.5

Evaluate the exponent.

$\frac{3}{2^{2}} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.4

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

$\frac{3}{4} - \cos^{2} (45) - \sin^{2} (30)$

Step 1.5

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

$\frac{3}{4} - {(\frac{\sqrt{2}}{2})}^{2} - \sin^{2} (30)$

Step 1.6

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

$\frac{3}{4} - \frac{{\sqrt{2}}^{2}}{2^{2}} - \sin^{2} (30)$

Step 1.7

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

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Step 1.7.1

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

$\frac{3}{4} - \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} - \sin^{2} (30)$

Step 1.7.2

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

$\frac{3}{4} - \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} - \sin^{2} (30)$

Step 1.7.3

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

$\frac{3}{4} - \frac{2^{\frac{2}{2}}}{2^{2}} - \sin^{2} (30)$

Step 1.7.4

Cancel the common factor of $2$ .

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Step 1.7.4.1

Cancel the common factor.

$\frac{3}{4} - \frac{2^{\frac{2}{2}}}{2^{2}} - \sin^{2} (30)$

Step 1.7.4.2

Rewrite the expression.

$\frac{3}{4} - \frac{2^{1}}{2^{2}} - \sin^{2} (30)$

Step 1.7.5

Evaluate the exponent.

$\frac{3}{4} - \frac{2}{2^{2}} - \sin^{2} (30)$

Step 1.8

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

$\frac{3}{4} - \frac{2}{4} - \sin^{2} (30)$

Step 1.9

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

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Step 1.9.1

Factor $2$ out of $2$ .

$\frac{3}{4} - \frac{2 (1)}{4} - \sin^{2} (30)$

Step 1.9.2

Cancel the common factors.

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Step 1.9.2.1

Factor $2$ out of $4$ .

$\frac{3}{4} - \frac{2 \cdot 1}{2 \cdot 2} - \sin^{2} (30)$

Step 1.9.2.2

Cancel the common factor.

$\frac{3}{4} - \frac{2 \cdot 1}{2 \cdot 2} - \sin^{2} (30)$

Step 1.9.2.3

Rewrite the expression.

$\frac{3}{4} - \frac{1}{2} - \sin^{2} (30)$

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

One to any power is one.

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

Step 1.13

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

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

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Subtract $1$ from $3$ .

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

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

Factor $2$ out of $2$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $2$ out of $4$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Move the negative in front of the fraction.

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

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 $1$ .

$\frac{0}{2}$

Step 4.2.2

Divide $0$ by $2$ .

$0$