Trigonometry Examples

$\sin (x) = \frac{\sqrt{6}}{7}$

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.

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

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.

$Adjacent = \sqrt{{(7)}^{2} - {(\sqrt{6})}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

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

Adjacent $= \sqrt{49 - {(\sqrt{6})}^{2}}$

Step 4.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

Adjacent $= \sqrt{49 - {(6^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

Adjacent $= \sqrt{49 - 6^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

Adjacent $= \sqrt{49 - 6^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

Adjacent $= \sqrt{49 - 6^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

Adjacent $= \sqrt{49 - 6}$

Step 4.2.5

Evaluate the exponent.

Adjacent $= \sqrt{49 - 1 \cdot 6}$

Step 4.3

Multiply $- 1$ by $6$ .

Adjacent $= \sqrt{49 - 6}$

Step 4.4

Subtract $6$ from $49$ .

Adjacent $= \sqrt{43}$

Step 5

Find the value of cosine.

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Step 5.1

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

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

Step 5.2

Substitute in the known values.

$\cos (x) = \frac{\sqrt{43}}{7}$

Step 6

Find the value of tangent.

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Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\tan (x) = \frac{\sqrt{6}}{\sqrt{43}}$

Step 6.3

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

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Step 6.3.1

Multiply $\frac{\sqrt{6}}{\sqrt{43}}$ by $\frac{\sqrt{43}}{\sqrt{43}}$ .

$\tan (x) = \frac{\sqrt{6}}{\sqrt{43}} \cdot \frac{\sqrt{43}}{\sqrt{43}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply $\frac{\sqrt{6}}{\sqrt{43}}$ by $\frac{\sqrt{43}}{\sqrt{43}}$ .

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{\sqrt{43} \sqrt{43}}$

Step 6.3.2.2

Raise $\sqrt{43}$ to the power of $1$ .

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{\sqrt{43} \sqrt{43}}$

Step 6.3.2.3

Raise $\sqrt{43}$ to the power of $1$ .

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{\sqrt{43} \sqrt{43}}$

Step 6.3.2.4

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

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{{\sqrt{43}}^{1 + 1}}$

Step 6.3.2.5

Add $1$ and $1$ .

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{{\sqrt{43}}^{2}}$

Step 6.3.2.6

Rewrite ${\sqrt{43}}^{2}$ as $43$ .

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Step 6.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{43}$ as $43^{\frac{1}{2}}$ .

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{{(43^{\frac{1}{2}})}^{2}}$

Step 6.3.2.6.2

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

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{43^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.6.3

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

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{43^{\frac{2}{2}}}$

Step 6.3.2.6.4

Cancel the common factor of $2$ .

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Step 6.3.2.6.4.1

Cancel the common factor.

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{43^{\frac{2}{2}}}$

Step 6.3.2.6.4.2

Rewrite the expression.

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{43}$

Step 6.3.2.6.5

Evaluate the exponent.

$\tan (x) = \frac{\sqrt{6} \sqrt{43}}{43}$

Step 6.3.3

Simplify the numerator.

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Step 6.3.3.1

Combine using the product rule for radicals.

$\tan (x) = \frac{\sqrt{6 \cdot 43}}{43}$

Step 6.3.3.2

Multiply $6$ by $43$ .

$\tan (x) = \frac{\sqrt{258}}{43}$

Step 7

Find the value of cotangent.

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Step 7.1

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

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

Step 7.2

Substitute in the known values.

$\cot (x) = \frac{\sqrt{43}}{\sqrt{6}}$

Step 7.3

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

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Step 7.3.1

Multiply $\frac{\sqrt{43}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\cot (x) = \frac{\sqrt{43}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 7.3.2

Combine and simplify the denominator.

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Step 7.3.2.1

Multiply $\frac{\sqrt{43}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 7.3.2.2

Raise $\sqrt{6}$ to the power of $1$ .

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 7.3.2.3

Raise $\sqrt{6}$ to the power of $1$ .

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 7.3.2.4

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

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 7.3.2.5

Add $1$ and $1$ .

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 7.3.2.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 7.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 7.3.2.6.2

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

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 7.3.2.6.3

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

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{6^{\frac{2}{2}}}$

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.

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 7.3.2.6.4.2

Rewrite the expression.

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{6}$

Step 7.3.2.6.5

Evaluate the exponent.

$\cot (x) = \frac{\sqrt{43} \sqrt{6}}{6}$

Step 7.3.3

Simplify the numerator.

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Step 7.3.3.1

Combine using the product rule for radicals.

$\cot (x) = \frac{\sqrt{43 \cdot 6}}{6}$

Step 7.3.3.2

Multiply $43$ by $6$ .

$\cot (x) = \frac{\sqrt{258}}{6}$

Step 8

Find the value of secant.

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Step 8.1

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

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

Step 8.2

Substitute in the known values.

$\sec (x) = \frac{7}{\sqrt{43}}$

Step 8.3

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

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Step 8.3.1

Multiply $\frac{7}{\sqrt{43}}$ by $\frac{\sqrt{43}}{\sqrt{43}}$ .

$\sec (x) = \frac{7}{\sqrt{43}} \cdot \frac{\sqrt{43}}{\sqrt{43}}$

Step 8.3.2

Combine and simplify the denominator.

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Step 8.3.2.1

Multiply $\frac{7}{\sqrt{43}}$ by $\frac{\sqrt{43}}{\sqrt{43}}$ .

$\sec (x) = \frac{7 \sqrt{43}}{\sqrt{43} \sqrt{43}}$

Step 8.3.2.2

Raise $\sqrt{43}$ to the power of $1$ .

$\sec (x) = \frac{7 \sqrt{43}}{\sqrt{43} \sqrt{43}}$

Step 8.3.2.3

Raise $\sqrt{43}$ to the power of $1$ .

$\sec (x) = \frac{7 \sqrt{43}}{\sqrt{43} \sqrt{43}}$

Step 8.3.2.4

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

$\sec (x) = \frac{7 \sqrt{43}}{{\sqrt{43}}^{1 + 1}}$

Step 8.3.2.5

Add $1$ and $1$ .

$\sec (x) = \frac{7 \sqrt{43}}{{\sqrt{43}}^{2}}$

Step 8.3.2.6

Rewrite ${\sqrt{43}}^{2}$ as $43$ .

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Step 8.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{43}$ as $43^{\frac{1}{2}}$ .

$\sec (x) = \frac{7 \sqrt{43}}{{(43^{\frac{1}{2}})}^{2}}$

Step 8.3.2.6.2

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

$\sec (x) = \frac{7 \sqrt{43}}{43^{\frac{1}{2} \cdot 2}}$

Step 8.3.2.6.3

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

$\sec (x) = \frac{7 \sqrt{43}}{43^{\frac{2}{2}}}$

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.

$\sec (x) = \frac{7 \sqrt{43}}{43^{\frac{2}{2}}}$

Step 8.3.2.6.4.2

Rewrite the expression.

$\sec (x) = \frac{7 \sqrt{43}}{43}$

Step 8.3.2.6.5

Evaluate the exponent.

$\sec (x) = \frac{7 \sqrt{43}}{43}$

Step 9

Find the value of cosecant.

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Step 9.1

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

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

Step 9.2

Substitute in the known values.

$\csc (x) = \frac{7}{\sqrt{6}}$

Step 9.3

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

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Step 9.3.1

Multiply $\frac{7}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\csc (x) = \frac{7}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 9.3.2

Combine and simplify the denominator.

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Step 9.3.2.1

Multiply $\frac{7}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\csc (x) = \frac{7 \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 9.3.2.2

Raise $\sqrt{6}$ to the power of $1$ .

$\csc (x) = \frac{7 \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 9.3.2.3

Raise $\sqrt{6}$ to the power of $1$ .

$\csc (x) = \frac{7 \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 9.3.2.4

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

$\csc (x) = \frac{7 \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 9.3.2.5

Add $1$ and $1$ .

$\csc (x) = \frac{7 \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 9.3.2.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 9.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\csc (x) = \frac{7 \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 9.3.2.6.2

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

$\csc (x) = \frac{7 \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 9.3.2.6.3

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

$\csc (x) = \frac{7 \sqrt{6}}{6^{\frac{2}{2}}}$

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.

$\csc (x) = \frac{7 \sqrt{6}}{6^{\frac{2}{2}}}$

Step 9.3.2.6.4.2

Rewrite the expression.

$\csc (x) = \frac{7 \sqrt{6}}{6}$

Step 9.3.2.6.5

Evaluate the exponent.

$\csc (x) = \frac{7 \sqrt{6}}{6}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{\sqrt{6}}{7}$

$\cos (x) = \frac{\sqrt{43}}{7}$

$\tan (x) = \frac{\sqrt{258}}{43}$

$\cot (x) = \frac{\sqrt{258}}{6}$

$\sec (x) = \frac{7 \sqrt{43}}{43}$

$\csc (x) = \frac{7 \sqrt{6}}{6}$