Trigonometry Examples

$\csc (\arccos (\frac{1}{2})) - \cos (\arctan (- \frac{8}{15}))$

Step 1

Draw a triangle in the plane with vertices $(\frac{1}{2}, \sqrt{1^{2} - {(\frac{1}{2})}^{2}})$ , $(\frac{1}{2}, 0)$ , and the origin. Then $\arccos (\frac{1}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{1}{2}, \sqrt{1^{2} - {(\frac{1}{2})}^{2}})$ . Therefore, $\csc (\arccos (\frac{1}{2}))$ is $\frac{2 \sqrt{3}}{3}$ .

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Step 1.1

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

$\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.2

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.3

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.4

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.5

Add $1$ and $1$ .

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} - \cos (\arctan (- \frac{8}{15}))$

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 $3^{\frac{1}{2}}$ .

$\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.6.2

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

$\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.6.3

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

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{3}}{3^{1}} - \cos (\arctan (- \frac{8}{15}))$

Step 1.6.5

Evaluate the exponent.

$\frac{2 \sqrt{3}}{3} - \cos (\arctan (- \frac{8}{15}))$

Step 2

Draw a triangle in the plane with vertices $(1, - \frac{8}{15})$ , $(1, 0)$ , and the origin. Then $\arctan (- \frac{8}{15})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, - \frac{8}{15})$ . Therefore, $\cos (\arctan (- \frac{8}{15}))$ is $\frac{15}{17}$ .

$\frac{2 \sqrt{3}}{3} - \frac{15}{17}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{3} - \frac{15}{17}$

Decimal Form:

$0.27234759 \dots$