Trigonometry Examples

$\sec (x) = - \frac{5}{2}$ , $\tan (x) < 0$

Step 1

The tangent function is negative in the second and fourth quadrants. The secant function is negative in the second and third quadrants. The set of solutions for $x$ are limited to the second quadrant since that is the only quadrant found in both sets.

Solution is in the second quadrant.

Step 2

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

$\sec (x) = \frac{hypotenuse}{adjacent}$

Step 3

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

$Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Step 4

Replace the known values in the equation.

$Opposite = \sqrt{{(5)}^{2} - {(- 2)}^{2}}$

Step 5

Simplify inside the radical.

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Step 5.1

Raise $5$ to the power of $2$ .

Opposite $= \sqrt{25 - {(- 2)}^{2}}$

Step 5.2

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

Opposite $= \sqrt{25 - 1 \cdot 4}$

Step 5.3

Multiply $- 1$ by $4$ .

Opposite $= \sqrt{25 - 4}$

Step 5.4

Subtract $4$ from $25$ .

Opposite $= \sqrt{21}$

Step 6

Find the value of sine.

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Step 6.1

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

$\sin (x) = \frac{opp}{hyp}$

Step 6.2

Substitute in the known values.

$\sin (x) = \frac{\sqrt{21}}{5}$

Step 7

Find the value of cosine.

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Step 7.1

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

$\cos (x) = \frac{adj}{hyp}$

Step 7.2

Substitute in the known values.

$\cos (x) = \frac{- 2}{5}$

Step 7.3

Move the negative in front of the fraction.

$\cos (x) = - \frac{2}{5}$

Step 8

Find the value of tangent.

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Step 8.1

Use the definition of tangent to find the value of $\tan (x)$ .

$\tan (x) = \frac{opp}{adj}$

Step 8.2

Substitute in the known values.

$\tan (x) = \frac{\sqrt{21}}{- 2}$

Step 8.3

Move the negative in front of the fraction.

$\tan (x) = - \frac{\sqrt{21}}{2}$

Step 9

Find the value of cotangent.

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Step 9.1

Use the definition of cotangent to find the value of $\cot (x)$ .

$\cot (x) = \frac{adj}{opp}$

Step 9.2

Substitute in the known values.

$\cot (x) = \frac{- 2}{\sqrt{21}}$

Step 9.3

Simplify the value of $\cot (x)$ .

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Step 9.3.1

Move the negative in front of the fraction.

$\cot (x) = - \frac{2}{\sqrt{21}}$

Step 9.3.2

Multiply $\frac{2}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\cot (x) = - (\frac{2}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}})$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{2}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\cot (x) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 9.3.3.2

Raise $\sqrt{21}$ to the power of $1$ .

$\cot (x) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 9.3.3.3

Raise $\sqrt{21}$ to the power of $1$ .

$\cot (x) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 9.3.3.4

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

$\cot (x) = - \frac{2 \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\cot (x) = - \frac{2 \sqrt{21}}{{\sqrt{21}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\cot (x) = - \frac{2 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\cot (x) = - \frac{2 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\cot (x) = - \frac{2 \sqrt{21}}{21^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

$\cot (x) = - \frac{2 \sqrt{21}}{21^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\cot (x) = - \frac{2 \sqrt{21}}{21}$

Step 9.3.3.6.5

Evaluate the exponent.

$\cot (x) = - \frac{2 \sqrt{21}}{21}$

Step 10

Find the value of cosecant.

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Step 10.1

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

$\csc (x) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

$\csc (x) = \frac{5}{\sqrt{21}}$

Step 10.3

Simplify the value of $\csc (x)$ .

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Step 10.3.1

Multiply $\frac{5}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\csc (x) = \frac{5}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}}$

Step 10.3.2

Combine and simplify the denominator.

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Step 10.3.2.1

Multiply $\frac{5}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\csc (x) = \frac{5 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 10.3.2.2

Raise $\sqrt{21}$ to the power of $1$ .

$\csc (x) = \frac{5 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 10.3.2.3

Raise $\sqrt{21}$ to the power of $1$ .

$\csc (x) = \frac{5 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 10.3.2.4

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

$\csc (x) = \frac{5 \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Step 10.3.2.5

Add $1$ and $1$ .

$\csc (x) = \frac{5 \sqrt{21}}{{\sqrt{21}}^{2}}$

Step 10.3.2.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 10.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\csc (x) = \frac{5 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

Step 10.3.2.6.2

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

$\csc (x) = \frac{5 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

Step 10.3.2.6.3

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

$\csc (x) = \frac{5 \sqrt{21}}{21^{\frac{2}{2}}}$

Step 10.3.2.6.4

Cancel the common factor of $2$ .

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Step 10.3.2.6.4.1

Cancel the common factor.

$\csc (x) = \frac{5 \sqrt{21}}{21^{\frac{2}{2}}}$

Step 10.3.2.6.4.2

Rewrite the expression.

$\csc (x) = \frac{5 \sqrt{21}}{21}$

Step 10.3.2.6.5

Evaluate the exponent.

$\csc (x) = \frac{5 \sqrt{21}}{21}$

Step 11

This is the solution to each trig value.

$\sin (x) = \frac{\sqrt{21}}{5}$

$\cos (x) = - \frac{2}{5}$

$\tan (x) = - \frac{\sqrt{21}}{2}$

$\cot (x) = - \frac{2 \sqrt{21}}{21}$

$\sec (x) = - \frac{5}{2}$

$\csc (x) = \frac{5 \sqrt{21}}{21}$