Trigonometry Examples

$\tan (θ) = 2$ , $\sin (θ) < 0$

Step 1

The sine function is negative in the third and fourth quadrants. The tangent function is positive in the first and third quadrants. The set of solutions for $θ$ are limited to the third quadrant since that is the only quadrant found in both sets.

Solution is in the third quadrant.

Step 2

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 3

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 4

Replace the known values in the equation.

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

Step 5

Simplify inside the radical.

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Step 5.1

Raise $- 2$ to the power of $2$ .

Hypotenuse $= \sqrt{4 + {(- 1)}^{2}}$

Step 5.2

Raise $- 1$ to the power of $2$ .

Hypotenuse $= \sqrt{4 + 1}$

Step 5.3

Add $4$ and $1$ .

Hypotenuse $= \sqrt{5}$

Step 6

Find the value of sine.

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Step 6.1

Use the definition of sine to find the value of $\sin (θ)$ .

$\sin (θ) = \frac{opp}{hyp}$

Step 6.2

Substitute in the known values.

$\sin (θ) = \frac{- 2}{\sqrt{5}}$

Step 6.3

Simplify the value of $\sin (θ)$ .

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Step 6.3.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{2}{\sqrt{5}}$

Step 6.3.2

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = - (\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = - \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.3.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (θ) = - \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.3.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (θ) = - \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.3.3.4

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

$\sin (θ) = - \frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 6.3.3.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{2 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 6.3.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 6.3.3.6.2

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

$\sin (θ) = - \frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.6.3

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

$\sin (θ) = - \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

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.

$\sin (θ) = - \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.3.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{2 \sqrt{5}}{5}$

Step 6.3.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{2 \sqrt{5}}{5}$

Step 7

Find the value of cosine.

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Step 7.1

Use the definition of cosine to find the value of $\cos (θ)$ .

$\cos (θ) = \frac{adj}{hyp}$

Step 7.2

Substitute in the known values.

$\cos (θ) = \frac{- 1}{\sqrt{5}}$

Step 7.3

Simplify the value of $\cos (θ)$ .

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Step 7.3.1

Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{\sqrt{5}}$

Step 7.3.2

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\cos (θ) = - (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\cos (θ) = - \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\cos (θ) = - \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\cos (θ) = - \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.3.4

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

$\cos (θ) = - \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 7.3.3.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 7.3.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 7.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 7.3.3.6.2

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

$\cos (θ) = - \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 7.3.3.6.3

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

$\cos (θ) = - \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.3.3.6.4

Cancel the common factor of $2$ .

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Step 7.3.3.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.3.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{5}}{5}$

Step 7.3.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{5}}{5}$

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.

$\cot (θ) = \frac{- 1}{- 2}$

Step 8.3

Dividing two negative values results in a positive value.

$\cot (θ) = \frac{1}{2}$

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.

$\sec (θ) = \frac{\sqrt{5}}{- 1}$

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 $\frac{\sqrt{5}}{- 1}$ .

$\sec (θ) = - 1 \cdot \sqrt{5}$

Step 9.3.2

Rewrite $- 1 \cdot \sqrt{5}$ as $- \sqrt{5}$ .

$\sec (θ) = - \sqrt{5}$

Step 10

Find the value of cosecant.

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Step 10.1

Use the definition of cosecant to find the value of $\csc (θ)$ .

$\csc (θ) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

$\csc (θ) = \frac{\sqrt{5}}{- 2}$

Step 10.3

Move the negative in front of the fraction.

$\csc (θ) = - \frac{\sqrt{5}}{2}$

Step 11

This is the solution to each trig value.

$\sin (θ) = - \frac{2 \sqrt{5}}{5}$

$\cos (θ) = - \frac{\sqrt{5}}{5}$

$\tan (θ) = 2$

$\cot (θ) = \frac{1}{2}$

$\sec (θ) = - \sqrt{5}$

$\csc (θ) = - \frac{\sqrt{5}}{2}$