Trigonometry Examples

$\sin (θ) = \frac{\sqrt{5}}{7}$

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.

$\sin (θ) = \frac{opposite}{hypotenuse}$

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.

$Adjacent = \sqrt{{(7)}^{2} - {(\sqrt{5})}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise $7$ to the power of $2$ .

Adjacent $= \sqrt{49 - {(\sqrt{5})}^{2}}$

Step 4.2

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

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Step 4.2.1

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

Adjacent $= \sqrt{49 - {(5^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

Adjacent $= \sqrt{49 - 5^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

Adjacent $= \sqrt{49 - 5^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

Adjacent $= \sqrt{49 - 5^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

Adjacent $= \sqrt{49 - 5}$

Step 4.2.5

Evaluate the exponent.

Adjacent $= \sqrt{49 - 1 \cdot 5}$

Step 4.3

Multiply $- 1$ by $5$ .

Adjacent $= \sqrt{49 - 5}$

Step 4.4

Subtract $5$ from $49$ .

Adjacent $= \sqrt{44}$

Step 4.5

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 4.5.1

Factor $4$ out of $44$ .

Adjacent $= \sqrt{4 (11)}$

Step 4.5.2

Rewrite $4$ as $2^{2}$ .

Adjacent $= \sqrt{2^{2} \cdot 11}$

Step 4.6

Pull terms out from under the radical.

Adjacent $= 2 \sqrt{11}$

Step 5

Find the value of cosine.

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Step 5.1

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

$\cos (θ) = \frac{adj}{hyp}$

Step 5.2

Substitute in the known values.

$\cos (θ) = \frac{2 \sqrt{11}}{7}$

Step 6

Find the value of tangent.

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Step 6.1

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

$\tan (θ) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

$\tan (θ) = \frac{\sqrt{5}}{2 \sqrt{11}}$

Step 6.3

Simplify the value of $\tan (θ)$ .

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Step 6.3.1

Multiply $\frac{\sqrt{5}}{2 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\tan (θ) = \frac{\sqrt{5}}{2 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply $\frac{\sqrt{5}}{2 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 \sqrt{11} \sqrt{11}}$

Step 6.3.2.2

Move $\sqrt{11}$ .

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 (\sqrt{11} \sqrt{11})}$

Step 6.3.2.3

Raise $\sqrt{11}$ to the power of $1$ .

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 (\sqrt{11} \sqrt{11})}$

Step 6.3.2.4

Raise $\sqrt{11}$ to the power of $1$ .

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 (\sqrt{11} \sqrt{11})}$

Step 6.3.2.5

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

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 {\sqrt{11}}^{1 + 1}}$

Step 6.3.2.6

Add $1$ and $1$ .

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 {\sqrt{11}}^{2}}$

Step 6.3.2.7

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 6.3.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 {(11^{\frac{1}{2}})}^{2}}$

Step 6.3.2.7.2

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

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 \cdot 11^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.7.3

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

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 \cdot 11^{\frac{2}{2}}}$

Step 6.3.2.7.4

Cancel the common factor of $2$ .

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Step 6.3.2.7.4.1

Cancel the common factor.

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 \cdot 11^{\frac{2}{2}}}$

Step 6.3.2.7.4.2

Rewrite the expression.

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 \cdot 11}$

Step 6.3.2.7.5

Evaluate the exponent.

$\tan (θ) = \frac{\sqrt{5} \sqrt{11}}{2 \cdot 11}$

Step 6.3.3

Simplify the numerator.

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Step 6.3.3.1

Combine using the product rule for radicals.

$\tan (θ) = \frac{\sqrt{5 \cdot 11}}{2 \cdot 11}$

Step 6.3.3.2

Multiply $5$ by $11$ .

$\tan (θ) = \frac{\sqrt{55}}{2 \cdot 11}$

Step 6.3.4

Multiply $2$ by $11$ .

$\tan (θ) = \frac{\sqrt{55}}{22}$

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.

$\cot (θ) = \frac{2 \sqrt{11}}{\sqrt{5}}$

Step 7.3

Simplify the value of $\cot (θ)$ .

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Step 7.3.1

Multiply $\frac{2 \sqrt{11}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\cot (θ) = \frac{2 \sqrt{11}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 7.3.2

Combine and simplify the denominator.

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Step 7.3.2.1

Multiply $\frac{2 \sqrt{11}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.2.2

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

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.2.3

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

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.2.4

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

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 7.3.2.5

Add $1$ and $1$ .

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 7.3.2.6

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

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Step 7.3.2.6.1

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

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 7.3.2.6.2

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

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 7.3.2.6.3

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

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.3.2.6.4

Cancel the common factor of $2$ .

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Step 7.3.2.6.4.1

Cancel the common factor.

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.3.2.6.4.2

Rewrite the expression.

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{5}$

Step 7.3.2.6.5

Evaluate the exponent.

$\cot (θ) = \frac{2 \sqrt{11} \sqrt{5}}{5}$

Step 7.3.3

Simplify the numerator.

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Step 7.3.3.1

Combine using the product rule for radicals.

$\cot (θ) = \frac{2 \sqrt{5 \cdot 11}}{5}$

Step 7.3.3.2

Multiply $5$ by $11$ .

$\cot (θ) = \frac{2 \sqrt{55}}{5}$

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.

$\sec (θ) = \frac{7}{2 \sqrt{11}}$

Step 8.3

Simplify the value of $\sec (θ)$ .

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Step 8.3.1

Multiply $\frac{7}{2 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\sec (θ) = \frac{7}{2 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 8.3.2

Combine and simplify the denominator.

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Step 8.3.2.1

Multiply $\frac{7}{2 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\sec (θ) = \frac{7 \sqrt{11}}{2 \sqrt{11} \sqrt{11}}$

Step 8.3.2.2

Move $\sqrt{11}$ .

$\sec (θ) = \frac{7 \sqrt{11}}{2 (\sqrt{11} \sqrt{11})}$

Step 8.3.2.3

Raise $\sqrt{11}$ to the power of $1$ .

$\sec (θ) = \frac{7 \sqrt{11}}{2 (\sqrt{11} \sqrt{11})}$

Step 8.3.2.4

Raise $\sqrt{11}$ to the power of $1$ .

$\sec (θ) = \frac{7 \sqrt{11}}{2 (\sqrt{11} \sqrt{11})}$

Step 8.3.2.5

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

$\sec (θ) = \frac{7 \sqrt{11}}{2 {\sqrt{11}}^{1 + 1}}$

Step 8.3.2.6

Add $1$ and $1$ .

$\sec (θ) = \frac{7 \sqrt{11}}{2 {\sqrt{11}}^{2}}$

Step 8.3.2.7

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 8.3.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sec (θ) = \frac{7 \sqrt{11}}{2 {(11^{\frac{1}{2}})}^{2}}$

Step 8.3.2.7.2

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

$\sec (θ) = \frac{7 \sqrt{11}}{2 \cdot 11^{\frac{1}{2} \cdot 2}}$

Step 8.3.2.7.3

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

$\sec (θ) = \frac{7 \sqrt{11}}{2 \cdot 11^{\frac{2}{2}}}$

Step 8.3.2.7.4

Cancel the common factor of $2$ .

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Step 8.3.2.7.4.1

Cancel the common factor.

$\sec (θ) = \frac{7 \sqrt{11}}{2 \cdot 11^{\frac{2}{2}}}$

Step 8.3.2.7.4.2

Rewrite the expression.

$\sec (θ) = \frac{7 \sqrt{11}}{2 \cdot 11}$

Step 8.3.2.7.5

Evaluate the exponent.

$\sec (θ) = \frac{7 \sqrt{11}}{2 \cdot 11}$

Step 8.3.3

Multiply $2$ by $11$ .

$\sec (θ) = \frac{7 \sqrt{11}}{22}$

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.

$\csc (θ) = \frac{7}{\sqrt{5}}$

Step 9.3

Simplify the value of $\csc (θ)$ .

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Step 9.3.1

Multiply $\frac{7}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\csc (θ) = \frac{7}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 9.3.2

Combine and simplify the denominator.

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Step 9.3.2.1

Multiply $\frac{7}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\csc (θ) = \frac{7 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 9.3.2.2

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

$\csc (θ) = \frac{7 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 9.3.2.3

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

$\csc (θ) = \frac{7 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 9.3.2.4

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

$\csc (θ) = \frac{7 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 9.3.2.5

Add $1$ and $1$ .

$\csc (θ) = \frac{7 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 9.3.2.6

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

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Step 9.3.2.6.1

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

$\csc (θ) = \frac{7 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 9.3.2.6.2

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

$\csc (θ) = \frac{7 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 9.3.2.6.3

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

$\csc (θ) = \frac{7 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 9.3.2.6.4

Cancel the common factor of $2$ .

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Step 9.3.2.6.4.1

Cancel the common factor.

$\csc (θ) = \frac{7 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 9.3.2.6.4.2

Rewrite the expression.

$\csc (θ) = \frac{7 \sqrt{5}}{5}$

Step 9.3.2.6.5

Evaluate the exponent.

$\csc (θ) = \frac{7 \sqrt{5}}{5}$

Step 10

This is the solution to each trig value.

$\sin (θ) = \frac{\sqrt{5}}{7}$

$\cos (θ) = \frac{2 \sqrt{11}}{7}$

$\tan (θ) = \frac{\sqrt{55}}{22}$

$\cot (θ) = \frac{2 \sqrt{55}}{5}$

$\sec (θ) = \frac{7 \sqrt{11}}{22}$

$\csc (θ) = \frac{7 \sqrt{5}}{5}$