Trigonometry Examples

$\csc (θ) = \sqrt{2}$

Step 1

Use the definition of cosecant to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\csc (θ) = \frac{hypotenuse}{opposite}$

Step 2

Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.

$Adjacent = - \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = - \sqrt{{(\sqrt{2})}^{2} - {(1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate

$\sqrt{{(\sqrt{2})}^{2} - {(1)}^{2}}$ .

Adjacent

$= - \sqrt{{(\sqrt{2})}^{2} - {(1)}^{2}}$

Step 4.2

Rewrite

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

$2$ .

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Step 4.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

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

Adjacent

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

Step 4.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

Adjacent

$= - \sqrt{2^{\frac{1}{2} \cdot 2} - {(1)}^{2}}$

Step 4.2.3

Combine

$\frac{1}{2}$ and

$2$ .

Adjacent

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

Step 4.2.4

Cancel the common factor of

$2$ .

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Step 4.2.4.1

Cancel the common factor.

Adjacent

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

Step 4.2.4.2

Rewrite the expression.

Adjacent

$= - \sqrt{2 - {(1)}^{2}}$

Adjacent

$= - \sqrt{2 - {(1)}^{2}}$

Step 4.2.5

Evaluate the exponent.

Adjacent

$= - \sqrt{2 - {(1)}^{2}}$

Adjacent

$= - \sqrt{2 - {(1)}^{2}}$

Step 4.3

One to any power is one.

Adjacent

$= - \sqrt{2 - 1 \cdot 1}$

Step 4.4

Multiply

$- 1$ by

$1$ .

Adjacent

$= - \sqrt{2 - 1}$

Step 4.5

Subtract

$1$ from

$2$ .

Adjacent

$= - \sqrt{1}$

Step 4.6

Any root of

$1$ is

$1$ .

Adjacent

$= - 1 \cdot 1$

Step 4.7

Multiply

$- 1$ by

$1$ .

Adjacent

$= - 1$

Adjacent

$= - 1$

Step 5

Find the value of sine.

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Step 5.1

Use the definition of sine to find the value of

$\sin (θ)$ .

$\sin (θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (θ) = \frac{1}{\sqrt{2}}$

Step 5.3

Simplify the value of

$\sin (θ)$ .

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Step 5.3.1

Multiply

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

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

$\sin (θ) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.3.2

Combine and simplify the denominator.

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Step 5.3.2.1

Multiply

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

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

$\sin (θ) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.3.2.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\sin (θ) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.3.2.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\sin (θ) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.3.2.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.3.2.5

Add

$1$ and

$1$ .

$\sin (θ) = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.3.2.6

Rewrite

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

$2$ .

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Step 5.3.2.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

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

$\sin (θ) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.3.2.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sin (θ) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sin (θ) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.3.2.6.4

Cancel the common factor of

$2$ .

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Step 5.3.2.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.3.2.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{\sqrt{2}}{2}$

Step 5.3.2.6.5

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{2}}{2}$

Step 6

Find the value of cosine.

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Step 6.1

Use the definition of cosine to find the value of

$\cos (θ)$ .

$\cos (θ) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (θ) = \frac{- 1}{\sqrt{2}}$

Step 6.3

Simplify the value of

$\cos (θ)$ .

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Step 6.3.1

Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{\sqrt{2}}$

Step 6.3.2

Multiply

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

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

$\cos (θ) = - (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

Multiply

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

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

$\cos (θ) = - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.3.3.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\cos (θ) = - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.3.3.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\cos (θ) = - \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.3.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (θ) = - \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.3.3.5

Add

$1$ and

$1$ .

$\cos (θ) = - \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6.3.3.6

Rewrite

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

$2$ .

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Step 6.3.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

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

$\cos (θ) = - \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.3.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\cos (θ) = - \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\cos (θ) = - \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.3.3.6.4

Cancel the common factor of

$2$ .

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Step 6.3.3.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.3.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{2}}{2}$

Step 6.3.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{2}}{2}$

Step 7

Find the value of tangent.

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Step 7.1

Use the definition of tangent to find the value of

$\tan (θ)$ .

$\tan (θ) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

$\tan (θ) = \frac{1}{- 1}$

Step 7.3

Divide

$1$ by

$- 1$ .

$\tan (θ) = - 1$

Step 8

Find the value of cotangent.

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Step 8.1

Use the definition of cotangent to find the value of

$\cot (θ)$ .

$\cot (θ) = \frac{adj}{opp}$

Step 8.2

Substitute in the known values.

$\cot (θ) = \frac{- 1}{1}$

Step 8.3

Divide

$- 1$ by

$1$ .

$\cot (θ) = - 1$

Step 9

Find the value of secant.

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Step 9.1

Use the definition of secant to find the value of

$\sec (θ)$ .

$\sec (θ) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

$\sec (θ) = \frac{\sqrt{2}}{- 1}$

Step 9.3

Simplify the value of

$\sec (θ)$ .

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Step 9.3.1

Move the negative one from the denominator of

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

$\sec (θ) = - 1 \cdot \sqrt{2}$

Step 9.3.2

Rewrite

$- 1 \cdot \sqrt{2}$ as

$- \sqrt{2}$ .

$\sec (θ) = - \sqrt{2}$

Step 10

This is the solution to each trig value.

$\sin (θ) = \frac{\sqrt{2}}{2}$

$\cos (θ) = - \frac{\sqrt{2}}{2}$

$\tan (θ) = - 1$

$\cot (θ) = - 1$

$\sec (θ) = - \sqrt{2}$

$\csc (θ) = \sqrt{2}$