Trigonometry Examples

$- 1 + \sin^{2} (30) + \sin^{2} (60)$

Step 1

Simplify with factoring out.

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Step 1.1

Rewrite $- 1$ as $- 1 (1)$ .

$- 1 (1) + \sin^{2} (30) + \sin^{2} (60)$

Step 1.2

Factor $- 1$ out of $\sin^{2} (30)$ .

$- 1 (1) - 1 (- \sin^{2} (30)) + \sin^{2} (60)$

Step 1.3

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (30))$ .

$- 1 (1 - \sin^{2} (30)) + \sin^{2} (60)$

Step 1.4

Rewrite $- 1 (1 - \sin^{2} (30))$ as $- (1 - \sin^{2} (30))$ .

$- (1 - \sin^{2} (30)) + \sin^{2} (60)$

Step 2

Apply pythagorean identity.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$- {(\frac{\sqrt{3}}{2})}^{2} + \sin^{2} (60)$

Step 3.1.2

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

$- \frac{{\sqrt{3}}^{2}}{2^{2}} + \sin^{2} (60)$

Step 3.1.3

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

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Step 3.1.3.1

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

$- \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} + \sin^{2} (60)$

Step 3.1.3.2

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

$- \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} + \sin^{2} (60)$

Step 3.1.3.3

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

$- \frac{3^{\frac{2}{2}}}{2^{2}} + \sin^{2} (60)$

Step 3.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.3.4.1

Cancel the common factor.

$- \frac{3^{\frac{2}{2}}}{2^{2}} + \sin^{2} (60)$

Step 3.1.3.4.2

Rewrite the expression.

$- \frac{3^{1}}{2^{2}} + \sin^{2} (60)$

Step 3.1.3.5

Evaluate the exponent.

$- \frac{3}{2^{2}} + \sin^{2} (60)$

Step 3.1.4

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

$- \frac{3}{4} + \sin^{2} (60)$

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.1.7

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

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Step 3.1.7.1

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

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

Step 3.1.7.2

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

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

Step 3.1.7.3

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

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

Step 3.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.7.4.1

Cancel the common factor.

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

Step 3.1.7.4.2

Rewrite the expression.

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

Step 3.1.7.5

Evaluate the exponent.

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

Step 3.1.8

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

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

Step 3.2

Combine fractions.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Add $- 3$ and $3$ .

$\frac{0}{4}$

Step 3.2.2.2

Divide $0$ by $4$ .

$0$