Trigonometry Examples

$\sin^{2} (240) - \cos^{2} (180) + \tan^{2} (60)$

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 sine is negative in the third quadrant.

${(- \sin (60))}^{2} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.2

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

${(- \frac{\sqrt{3}}{2})}^{2} - \cos^{2} (180) + \tan^{2} (60)$

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 $- \frac{\sqrt{3}}{2}$ .

${(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.3.2

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

${(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.4

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

$1 \frac{{\sqrt{3}}^{2}}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.5

Multiply $\frac{{\sqrt{3}}^{2}}{2^{2}}$ by $1$ .

$\frac{{\sqrt{3}}^{2}}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.6

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

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Step 1.6.1

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

$\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.6.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.6.3

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

$\frac{3^{\frac{2}{2}}}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.6.4.2

Rewrite the expression.

$\frac{3^{1}}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.6.5

Evaluate the exponent.

$\frac{3}{2^{2}} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.7

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

$\frac{3}{4} - \cos^{2} (180) + \tan^{2} (60)$

Step 1.8

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.

$\frac{3}{4} - {(- \cos (0))}^{2} + \tan^{2} (60)$

Step 1.9

The exact value of $\cos (0)$ is $1$ .

$\frac{3}{4} - {(- 1 \cdot 1)}^{2} + \tan^{2} (60)$

Step 1.10

Multiply $- 1$ by $1$ .

$\frac{3}{4} - {(- 1)}^{2} + \tan^{2} (60)$

Step 1.11

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

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Step 1.11.1

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

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Step 1.11.1.1

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

$\frac{3}{4} + {(- 1)}^{1} {(- 1)}^{2} + \tan^{2} (60)$

Step 1.11.1.2

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

$\frac{3}{4} + {(- 1)}^{1 + 2} + \tan^{2} (60)$

Step 1.11.2

Add $1$ and $2$ .

$\frac{3}{4} + {(- 1)}^{3} + \tan^{2} (60)$

Step 1.12

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

$\frac{3}{4} - 1 + \tan^{2} (60)$

Step 1.13

The exact value of $\tan (60)$ is $\sqrt{3}$ .

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

Step 1.14

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

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Step 1.14.1

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

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

Step 1.14.2

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

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

Step 1.14.3

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

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

Step 1.14.4

Cancel the common factor of $2$ .

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Step 1.14.4.1

Cancel the common factor.

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

Step 1.14.4.2

Rewrite the expression.

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

Step 1.14.5

Evaluate the exponent.

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

Step 2

Find the common denominator.

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Step 2.1

Write $- 1$ as a fraction with denominator $1$ .

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

Step 2.2

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

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

Step 2.3

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

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

Step 2.4

Write $3$ as a fraction with denominator $1$ .

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

Step 2.5

Multiply $\frac{3}{1}$ by $\frac{4}{4}$ .

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

Step 2.6

Multiply $\frac{3}{1}$ by $\frac{4}{4}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $- 1$ by $4$ .

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

Step 4.2

Multiply $3$ by $4$ .

$\frac{3 - 4 + 12}{4}$

Step 5

Simplify by adding and subtracting.

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Step 5.1

Subtract $4$ from $3$ .

$\frac{- 1 + 12}{4}$

Step 5.2

Add $- 1$ and $12$ .

$\frac{11}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{11}{4}$

Decimal Form:

$2.75$

Mixed Number Form:

$2 \frac{3}{4}$