Trigonometry Examples

$\sin (θ_{1}) = - \frac{10}{13}$

Step 1

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

$\sin (θ_{1}) = \frac{opposite}{hypotenuse}$

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.

$Adjacent = \sqrt{{(13)}^{2} - {(- 10)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise $13$ to the power of $2$ .

Adjacent $= \sqrt{169 - {(- 10)}^{2}}$

Step 4.2

Raise $- 10$ to the power of $2$ .

Adjacent $= \sqrt{169 - 1 \cdot 100}$

Step 4.3

Multiply $- 1$ by $100$ .

Adjacent $= \sqrt{169 - 100}$

Step 4.4

Subtract $100$ from $169$ .

Adjacent $= \sqrt{69}$

Step 5

Find the value of cosine.

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Step 5.1

Use the definition of cosine to find the value of $\cos (θ_{1})$ .

$\cos (θ_{1}) = \frac{adj}{hyp}$

Step 5.2

Substitute in the known values.

$\cos (θ_{1}) = \frac{\sqrt{69}}{13}$

Step 6

Find the value of tangent.

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Step 6.1

Use the definition of tangent to find the value of $\tan (θ_{1})$ .

$\tan (θ_{1}) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

$\tan (θ_{1}) = \frac{- 10}{\sqrt{69}}$

Step 6.3

Simplify the value of $\tan (θ_{1})$ .

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Step 6.3.1

Move the negative in front of the fraction.

$\tan (θ_{1}) = - \frac{10}{\sqrt{69}}$

Step 6.3.2

Multiply $\frac{10}{\sqrt{69}}$ by $\frac{\sqrt{69}}{\sqrt{69}}$ .

$\tan (θ_{1}) = - (\frac{10}{\sqrt{69}} \cdot \frac{\sqrt{69}}{\sqrt{69}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

Multiply $\frac{10}{\sqrt{69}}$ by $\frac{\sqrt{69}}{\sqrt{69}}$ .

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{\sqrt{69} \sqrt{69}}$

Step 6.3.3.2

Raise $\sqrt{69}$ to the power of $1$ .

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{\sqrt{69} \sqrt{69}}$

Step 6.3.3.3

Raise $\sqrt{69}$ to the power of $1$ .

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{\sqrt{69} \sqrt{69}}$

Step 6.3.3.4

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

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{{\sqrt{69}}^{1 + 1}}$

Step 6.3.3.5

Add $1$ and $1$ .

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{{\sqrt{69}}^{2}}$

Step 6.3.3.6

Rewrite ${\sqrt{69}}^{2}$ as $69$ .

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Step 6.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{69}$ as $69^{\frac{1}{2}}$ .

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{{(69^{\frac{1}{2}})}^{2}}$

Step 6.3.3.6.2

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

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{69^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.6.3

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

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{69^{\frac{2}{2}}}$

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.

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{69^{\frac{2}{2}}}$

Step 6.3.3.6.4.2

Rewrite the expression.

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{69}$

Step 6.3.3.6.5

Evaluate the exponent.

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{69}$

Step 7

Find the value of cotangent.

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Step 7.1

Use the definition of cotangent to find the value of $\cot (θ_{1})$ .

$\cot (θ_{1}) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (θ_{1}) = \frac{\sqrt{69}}{- 10}$

Step 7.3

Move the negative in front of the fraction.

$\cot (θ_{1}) = - \frac{\sqrt{69}}{10}$

Step 8

Find the value of secant.

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Step 8.1

Use the definition of secant to find the value of $\sec (θ_{1})$ .

$\sec (θ_{1}) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (θ_{1}) = \frac{13}{\sqrt{69}}$

Step 8.3

Simplify the value of $\sec (θ_{1})$ .

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Step 8.3.1

Multiply $\frac{13}{\sqrt{69}}$ by $\frac{\sqrt{69}}{\sqrt{69}}$ .

$\sec (θ_{1}) = \frac{13}{\sqrt{69}} \cdot \frac{\sqrt{69}}{\sqrt{69}}$

Step 8.3.2

Combine and simplify the denominator.

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Step 8.3.2.1

Multiply $\frac{13}{\sqrt{69}}$ by $\frac{\sqrt{69}}{\sqrt{69}}$ .

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{\sqrt{69} \sqrt{69}}$

Step 8.3.2.2

Raise $\sqrt{69}$ to the power of $1$ .

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{\sqrt{69} \sqrt{69}}$

Step 8.3.2.3

Raise $\sqrt{69}$ to the power of $1$ .

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{\sqrt{69} \sqrt{69}}$

Step 8.3.2.4

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

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{{\sqrt{69}}^{1 + 1}}$

Step 8.3.2.5

Add $1$ and $1$ .

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{{\sqrt{69}}^{2}}$

Step 8.3.2.6

Rewrite ${\sqrt{69}}^{2}$ as $69$ .

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Step 8.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{69}$ as $69^{\frac{1}{2}}$ .

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{{(69^{\frac{1}{2}})}^{2}}$

Step 8.3.2.6.2

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

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{69^{\frac{1}{2} \cdot 2}}$

Step 8.3.2.6.3

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

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{69^{\frac{2}{2}}}$

Step 8.3.2.6.4

Cancel the common factor of $2$ .

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Step 8.3.2.6.4.1

Cancel the common factor.

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{69^{\frac{2}{2}}}$

Step 8.3.2.6.4.2

Rewrite the expression.

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{69}$

Step 8.3.2.6.5

Evaluate the exponent.

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{69}$

Step 9

Find the value of cosecant.

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Step 9.1

Use the definition of cosecant to find the value of $\csc (θ_{1})$ .

$\csc (θ_{1}) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (θ_{1}) = \frac{13}{- 10}$

Step 9.3

Move the negative in front of the fraction.

$\csc (θ_{1}) = - \frac{13}{10}$

Step 10

This is the solution to each trig value.

$\sin (θ_{1}) = - \frac{10}{13}$

$\cos (θ_{1}) = \frac{\sqrt{69}}{13}$

$\tan (θ_{1}) = - \frac{10 \sqrt{69}}{69}$

$\cot (θ_{1}) = - \frac{\sqrt{69}}{10}$

$\sec (θ_{1}) = \frac{13 \sqrt{69}}{69}$

$\csc (θ_{1}) = - \frac{13}{10}$