Trigonometry Examples

tan (θ) = 3

$\tan (θ) = 3$

Step 1

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

$\tan (θ) = \frac{opposite}{adjacent}$

Step 2

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

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(- 3)}^{2} + {(- 1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise

$- 3$ to the power of

$2$ .

Hypotenuse

$= \sqrt{9 + {(- 1)}^{2}}$

Step 4.2

Raise

$- 1$ to the power of

$2$ .

Hypotenuse

$= \sqrt{9 + 1}$

Step 4.3

Add

$9$ and

$1$ .

Hypotenuse

$= \sqrt{10}$

Hypotenuse

$= \sqrt{10}$

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.

$\sin (θ) = \frac{- 3}{\sqrt{10}}$

Step 5.3

Simplify the value of

$\sin (θ)$ .

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Step 5.3.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{3}{\sqrt{10}}$

Step 5.3.2

Multiply

$\frac{3}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (θ) = - (\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 5.3.3

Combine and simplify the denominator.

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Step 5.3.3.1

Multiply

$\frac{3}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 5.3.3.2

Raise

$\sqrt{10}$ to the power of

$1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 5.3.3.3

Raise

$\sqrt{10}$ to the power of

$1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 5.3.3.4

Use the power rule

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

$\sin (θ) = - \frac{3 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 5.3.3.5

Add

$1$ and

$1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 5.3.3.6

Rewrite

${\sqrt{10}}^{2}$ as

$10$ .

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Step 5.3.3.6.1

Use

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

$\sqrt{10}$ as

$10^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 5.3.3.6.2

Apply the power rule and multiply exponents,

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

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 5.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.3.3.6.4

Cancel the common factor of

$2$ .

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Step 5.3.3.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.3.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

Step 5.3.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

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.

$\cos (θ) = \frac{- 1}{\sqrt{10}}$

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.

$\cos (θ) = - \frac{1}{\sqrt{10}}$

Step 6.3.2

Multiply

$\frac{1}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (θ) = - (\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

Multiply

$\frac{1}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 6.3.3.2

Raise

$\sqrt{10}$ to the power of

$1$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 6.3.3.3

Raise

$\sqrt{10}$ to the power of

$1$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 6.3.3.4

Use the power rule

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

$\cos (θ) = - \frac{\sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 6.3.3.5

Add

$1$ and

$1$ .

$\cos (θ) = - \frac{\sqrt{10}}{{\sqrt{10}}^{2}}$

Step 6.3.3.6

Rewrite

${\sqrt{10}}^{2}$ as

$10$ .

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Step 6.3.3.6.1

Use

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

$\sqrt{10}$ as

$10^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{\sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 6.3.3.6.2

Apply the power rule and multiply exponents,

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

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

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.

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

Step 6.3.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{10}}{10}$

Step 6.3.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{10}}{10}$

Step 7

Find the value of cotangent.

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Step 7.1

Use the definition of cotangent to find the value of

$\cot (θ)$ .

$\cot (θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (θ) = \frac{- 1}{- 3}$

Step 7.3

Dividing two negative values results in a positive value.

$\cot (θ) = \frac{1}{3}$

Step 8

Find the value of secant.

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Step 8.1

Use the definition of secant to find the value of

$\sec (θ)$ .

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (θ) = \frac{\sqrt{10}}{- 1}$

Step 8.3

Simplify the value of

$\sec (θ)$ .

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Step 8.3.1

Move the negative one from the denominator of

$\frac{\sqrt{10}}{- 1}$ .

$\sec (θ) = - 1 \cdot \sqrt{10}$

Step 8.3.2

Rewrite

$- 1 \cdot \sqrt{10}$ as

$- \sqrt{10}$ .

$\sec (θ) = - \sqrt{10}$

Step 9

Find the value of cosecant.

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Step 9.1

Use the definition of cosecant to find the value of

$\csc (θ)$ .

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (θ) = \frac{\sqrt{10}}{- 3}$

Step 9.3

Move the negative in front of the fraction.

$\csc (θ) = - \frac{\sqrt{10}}{3}$

Step 10

This is the solution to each trig value.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

$\cos (θ) = - \frac{\sqrt{10}}{10}$

$\tan (θ) = 3$

$\cot (θ) = \frac{1}{3}$

$\sec (θ) = - \sqrt{10}$

$\csc (θ) = - \frac{\sqrt{10}}{3}$