Trigonometry Examples

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

Step 1

Multiply the exponents in ${(\cos^{2} (30))}^{2}$ .

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Step 1.1

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

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

Step 1.2

Multiply $2$ by $2$ .

$\cos^{4} (30) \cdot 30 - \sin (- 60)$

Step 2

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

${(\frac{\sqrt{3}}{2})}^{4} \cdot 30 - \sin (- 60)$

Step 3

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

$\frac{{\sqrt{3}}^{4}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 4.1.1

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

$\frac{{(3^{\frac{1}{2}})}^{4}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4.1.2

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

$\frac{3^{\frac{1}{2} \cdot 4}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4.1.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{3^{\frac{4}{2}}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4.1.4

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

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Step 4.1.4.1

Factor $2$ out of $4$ .

$\frac{3^{\frac{2 \cdot 2}{2}}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4.1.4.2

Cancel the common factors.

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Step 4.1.4.2.1

Factor $2$ out of $2$ .

$\frac{3^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4.1.4.2.2

Cancel the common factor.

$\frac{3^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4.1.4.2.3

Rewrite the expression.

$\frac{3^{\frac{2}{1}}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4.1.4.2.4

Divide $2$ by $1$ .

$\frac{3^{2}}{2^{4}} \cdot 30 - \sin (- 60)$

Step 4.2

Raise $3$ to the power of $2$ .

$\frac{9}{2^{4}} \cdot 30 - \sin (- 60)$

Step 5

Raise $2$ to the power of $4$ .

$\frac{9}{16} \cdot 30 - \sin (- 60)$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Factor $2$ out of $16$ .

$\frac{9}{2 (8)} \cdot 30 - \sin (- 60)$

Step 6.2

Factor $2$ out of $30$ .

$\frac{9}{2 \cdot 8} \cdot (2 \cdot 15) - \sin (- 60)$

Step 6.3

Cancel the common factor.

$\frac{9}{2 \cdot 8} \cdot (2 \cdot 15) - \sin (- 60)$

Step 6.4

Rewrite the expression.

$\frac{9}{8} \cdot 15 - \sin (- 60)$

Step 7

Combine $\frac{9}{8}$ and $15$ .

$\frac{9 \cdot 15}{8} - \sin (- 60)$

Step 8

Multiply $9$ by $15$ .

$\frac{135}{8} - \sin (- 60)$

Step 9

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 fourth quadrant.

$\frac{135}{8} - - \sin (60)$

Step 10

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

$\frac{135}{8} - - \frac{\sqrt{3}}{2}$

Step 11

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

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Step 11.1

Multiply $- 1$ by $- 1$ .

$\frac{135}{8} + 1 \frac{\sqrt{3}}{2}$

Step 11.2

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

$\frac{135}{8} + \frac{\sqrt{3}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{135}{8} + \frac{\sqrt{3}}{2}$

Decimal Form:

$17.74102540 \dots$