Trigonometry Examples

$\sin (\frac{4 π}{3}) \cos (\frac{7 π}{6})$

Step 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.

$- \sin (\frac{π}{3}) \cos (\frac{7 π}{6})$

Step 2

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

$- \frac{\sqrt{3}}{2} \cos (\frac{7 π}{6})$

Step 3

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 third quadrant.

$- \frac{\sqrt{3}}{2} (- \cos (\frac{π}{6}))$

Step 4

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{3}}{2} (- \frac{\sqrt{3}}{2})$

Step 5

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

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Step 5.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{3}}{2} \frac{\sqrt{3}}{2}$

Step 5.2

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

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}$

Step 5.3

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

$\frac{\sqrt{3} \sqrt{3}}{2 \cdot 2}$

Step 5.4

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} \sqrt{3}}{2 \cdot 2}$

Step 5.5

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2 \cdot 2}$

Step 5.6

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

$\frac{{\sqrt{3}}^{1 + 1}}{2 \cdot 2}$

Step 5.7

Add $1$ and $1$ .

$\frac{{\sqrt{3}}^{2}}{2 \cdot 2}$

Step 5.8

Multiply $2$ by $2$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

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

Step 6.5

Evaluate the exponent.

$\frac{3}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$