Trigonometry Examples

$\csc (- 315)$

Step 1

Rewrite $- 315$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\csc (\frac{- 630}{2})$

Step 2

Apply the reciprocal identity to $\csc (\frac{- 630}{2})$ .

$\frac{1}{\sin (\frac{- 630}{2})}$

Step 3

Apply the sine half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (- 630)}{2}}}$

Step 4

Change the $\pm$ to $+$ because cosecant is positive in the first quadrant.

$\frac{1}{\sqrt{\frac{1 - \cos (- 630)}{2}}}$

Step 5

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (- 630)}{2}}}$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\frac{1}{\sqrt{\frac{1 - \cos (90)}{2}}}$

Step 5.1.2

The exact value of $\cos (90)$ is $0$ .

$\frac{1}{\sqrt{\frac{1 - 0}{2}}}$

Step 5.1.3

Multiply $- 1$ by $0$ .

$\frac{1}{\sqrt{\frac{1 + 0}{2}}}$

Step 5.1.4

Add $1$ and $0$ .

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

Step 5.2

Simplify the denominator.

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Step 5.2.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 5.2.2

Any root of $1$ is $1$ .

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

Step 5.2.3

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

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

Step 5.2.4

Combine and simplify the denominator.

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Step 5.2.4.1

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

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

Step 5.2.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.2.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.2.4.4

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

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

Step 5.2.4.5

Add $1$ and $1$ .

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

Step 5.2.4.6

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

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Step 5.2.4.6.1

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

$\frac{1}{\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}$

Step 5.2.4.6.2

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

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

Step 5.2.4.6.3

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

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

Step 5.2.4.6.4

Cancel the common factor of $2$ .

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Step 5.2.4.6.4.1

Cancel the common factor.

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

Step 5.2.4.6.4.2

Rewrite the expression.

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

Step 5.2.4.6.5

Evaluate the exponent.

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

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4

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

$\frac{2}{\sqrt{2}}$

Step 5.5

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

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

Step 5.6

Combine and simplify the denominator.

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Step 5.6.1

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

$\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.6.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.6.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.6.4

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

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

Step 5.6.5

Add $1$ and $1$ .

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

Step 5.6.6

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

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Step 5.6.6.1

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

$\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.6.6.2

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

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

Step 5.6.6.3

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

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

Step 5.6.6.4

Cancel the common factor of $2$ .

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Step 5.6.6.4.1

Cancel the common factor.

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

Step 5.6.6.4.2

Rewrite the expression.

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

Step 5.6.6.5

Evaluate the exponent.

$\frac{2 \sqrt{2}}{2}$

Step 5.7

Cancel the common factor of $2$ .

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Step 5.7.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2}$

Step 5.7.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$