Trigonometry Examples

$\tan (x) = \frac{5.6}{44}$

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.

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

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.

$Hypotenuse = \sqrt{{(- 5.6)}^{2} + {(- 44)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise $- 5.6$ to the power of $2$ .

Hypotenuse $= \sqrt{31.36 + {(- 44)}^{2}}$

Step 4.2

Raise $- 44$ to the power of $2$ .

Hypotenuse $= \sqrt{31.36 + 1936}$

Step 4.3

Add $31.36$ and $1936$ .

Hypotenuse $= \sqrt{1967.36}$

Step 5

Find the value of sine.

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Step 5.1

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

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

Step 5.2

Substitute in the known values.

$\sin (x) = \frac{- 5.6}{\sqrt{1967.36}}$

Step 5.3

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

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Step 5.3.1

Move the negative in front of the fraction.

$\sin (x) = - \frac{5.6}{\sqrt{1967.36}}$

Step 5.3.2

Multiply $\frac{5.6}{\sqrt{1967.36}}$ by $\frac{\sqrt{1967.36}}{\sqrt{1967.36}}$ .

$\sin (x) = - (\frac{5.6}{\sqrt{1967.36}} \cdot \frac{\sqrt{1967.36}}{\sqrt{1967.36}})$

Step 5.3.3

Combine and simplify the denominator.

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Step 5.3.3.1

Multiply $\frac{5.6}{\sqrt{1967.36}}$ by $\frac{\sqrt{1967.36}}{\sqrt{1967.36}}$ .

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{\sqrt{1967.36} \sqrt{1967.36}}$

Step 5.3.3.2

Raise $\sqrt{1967.36}$ to the power of $1$ .

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{\sqrt{1967.36} \sqrt{1967.36}}$

Step 5.3.3.3

Raise $\sqrt{1967.36}$ to the power of $1$ .

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{\sqrt{1967.36} \sqrt{1967.36}}$

Step 5.3.3.4

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

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{{\sqrt{1967.36}}^{1 + 1}}$

Step 5.3.3.5

Add $1$ and $1$ .

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{{\sqrt{1967.36}}^{2}}$

Step 5.3.3.6

Rewrite ${\sqrt{1967.36}}^{2}$ as $1967.36$ .

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Step 5.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1967.36}$ as ${1967.36}^{\frac{1}{2}}$ .

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{{({1967.36}^{\frac{1}{2}})}^{2}}$

Step 5.3.3.6.2

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

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{{1967.36}^{\frac{1}{2} \cdot 2}}$

Step 5.3.3.6.3

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

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{{1967.36}^{\frac{2}{2}}}$

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.

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{{1967.36}^{\frac{2}{2}}}$

Step 5.3.3.6.4.2

Rewrite the expression.

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{1967.36}$

Step 5.3.3.6.5

Evaluate the exponent.

$\sin (x) = - \frac{5.6 \sqrt{1967.36}}{1967.36}$

Step 5.3.4

Multiply $5.6$ by $\sqrt{1967.36}$ .

$\sin (x) = - \frac{248.38761965}{1967.36}$

Step 5.3.5

Simplify the expression.

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Step 5.3.5.1

Divide $248.38761965$ by $1967.36$ .

$\sin (x) = - 1 \cdot 0.12625427$

Step 5.3.5.2

Multiply $- 1$ by $0.12625427$ .

$\sin (x) = - 0.12625427$

Step 6

Find the value of cosine.

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Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\cos (x) = \frac{- 44}{\sqrt{1967.36}}$

Step 6.3

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

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Step 6.3.1

Move the negative in front of the fraction.

$\cos (x) = - \frac{44}{\sqrt{1967.36}}$

Step 6.3.2

Multiply $\frac{44}{\sqrt{1967.36}}$ by $\frac{\sqrt{1967.36}}{\sqrt{1967.36}}$ .

$\cos (x) = - (\frac{44}{\sqrt{1967.36}} \cdot \frac{\sqrt{1967.36}}{\sqrt{1967.36}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

Multiply $\frac{44}{\sqrt{1967.36}}$ by $\frac{\sqrt{1967.36}}{\sqrt{1967.36}}$ .

$\cos (x) = - \frac{44 \sqrt{1967.36}}{\sqrt{1967.36} \sqrt{1967.36}}$

Step 6.3.3.2

Raise $\sqrt{1967.36}$ to the power of $1$ .

$\cos (x) = - \frac{44 \sqrt{1967.36}}{\sqrt{1967.36} \sqrt{1967.36}}$

Step 6.3.3.3

Raise $\sqrt{1967.36}$ to the power of $1$ .

$\cos (x) = - \frac{44 \sqrt{1967.36}}{\sqrt{1967.36} \sqrt{1967.36}}$

Step 6.3.3.4

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

$\cos (x) = - \frac{44 \sqrt{1967.36}}{{\sqrt{1967.36}}^{1 + 1}}$

Step 6.3.3.5

Add $1$ and $1$ .

$\cos (x) = - \frac{44 \sqrt{1967.36}}{{\sqrt{1967.36}}^{2}}$

Step 6.3.3.6

Rewrite ${\sqrt{1967.36}}^{2}$ as $1967.36$ .

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Step 6.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1967.36}$ as ${1967.36}^{\frac{1}{2}}$ .

$\cos (x) = - \frac{44 \sqrt{1967.36}}{{({1967.36}^{\frac{1}{2}})}^{2}}$

Step 6.3.3.6.2

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

$\cos (x) = - \frac{44 \sqrt{1967.36}}{{1967.36}^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.6.3

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

$\cos (x) = - \frac{44 \sqrt{1967.36}}{{1967.36}^{\frac{2}{2}}}$

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.

$\cos (x) = - \frac{44 \sqrt{1967.36}}{{1967.36}^{\frac{2}{2}}}$

Step 6.3.3.6.4.2

Rewrite the expression.

$\cos (x) = - \frac{44 \sqrt{1967.36}}{1967.36}$

Step 6.3.3.6.5

Evaluate the exponent.

$\cos (x) = - \frac{44 \sqrt{1967.36}}{1967.36}$

Step 6.3.4

Evaluate the root.

$\cos (x) = - \frac{44 \cdot 44.35493208}{1967.36}$

Step 6.3.5

Multiply $44$ by $44.35493208$ .

$\cos (x) = - \frac{1951.6170116}{1967.36}$

Step 6.3.6

Divide $1951.6170116$ by $1967.36$ .

$\cos (x) = - 1 \cdot 0.99199791$

Step 6.3.7

Multiply $- 1$ by $0.99199791$ .

$\cos (x) = - 0.99199791$

Step 7

Divide $5.6$ by $44$ .

$\tan (x) = 0.1\overline{27}$

Step 8

Find the value of cotangent.

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Step 8.1

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

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

Step 8.2

Substitute in the known values.

$\cot (x) = \frac{- 44}{- 5.6}$

Step 8.3

Divide $- 44$ by $- 5.6$ .

$\cot (x) = 7.\overline{857142}$

Step 9

Find the value of secant.

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Step 9.1

Use the definition of secant to find the value of $\sec (x)$ .

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

Step 9.2

Substitute in the known values.

$\sec (x) = \frac{\sqrt{1967.36}}{- 44}$

Step 9.3

Move the negative in front of the fraction.

$\sec (x) = - \frac{\sqrt{1967.36}}{44}$

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.

$\csc (x) = \frac{\sqrt{1967.36}}{- 5.6}$

Step 10.3

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

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Step 10.3.1

Move the negative in front of the fraction.

$\csc (x) = - \frac{\sqrt{1967.36}}{5.6}$

Step 10.3.2

Evaluate the root.

$\csc (x) = - \frac{44.35493208}{5.6}$

Step 10.3.3

Divide $44.35493208$ by $5.6$ .

$\csc (x) = - 1 \cdot 7.92052358$

Step 10.3.4

Multiply $- 1$ by $7.92052358$ .

$\csc (x) = - 7.92052358$

Step 11

This is the solution to each trig value.

$\sin (x) = - 0.12625427$

$\cos (x) = - 0.99199791$

$\tan (x) = 0.1\overline{27}$

$\cot (x) = 7.\overline{857142}$

$\sec (x) = - \frac{\sqrt{1967.36}}{44}$

$\csc (x) = - 7.92052358$