Trigonometry Examples

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

Step 1

Simplify $\sqrt{1 - {(\frac{\sqrt{7}}{6})}^{2}}$ .

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Step 1.1

Apply the product rule to $\frac{\sqrt{7}}{6}$ .

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

Step 1.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Evaluate the exponent.

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

Step 1.3

Simplify the expression.

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Step 1.3.1

Raise $6$ to the power of $2$ .

$\cos (x) = \sqrt{1 - \frac{7}{36}}$

Step 1.3.2

Write $1$ as a fraction with a common denominator.

$\cos (x) = \sqrt{\frac{36}{36} - \frac{7}{36}}$

Step 1.3.3

Combine the numerators over the common denominator.

$\cos (x) = \sqrt{\frac{36 - 7}{36}}$

Step 1.3.4

Subtract $7$ from $36$ .

$\cos (x) = \sqrt{\frac{29}{36}}$

Step 1.4

Rewrite $\sqrt{\frac{29}{36}}$ as $\frac{\sqrt{29}}{\sqrt{36}}$ .

$\cos (x) = \frac{\sqrt{29}}{\sqrt{36}}$

Step 1.5

Simplify the denominator.

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Step 1.5.1

Rewrite $36$ as $6^{2}$ .

$\cos (x) = \frac{\sqrt{29}}{\sqrt{6^{2}}}$

Step 1.5.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (x) = \frac{\sqrt{29}}{6}$

Step 2

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (\frac{\sqrt{29}}{6})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (\frac{\sqrt{29}}{6})$ .

$x = 0.45666638$

Step 4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 (3.14159265) - 0.45666638$

Step 5

Simplify $2 (3.14159265) - 0.45666638$ .

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Step 5.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.45666638$

Step 5.2

Subtract $0.45666638$ from $6.2831853$ .

$x = 5.82651892$

Step 6

Find the period of $\cos (x)$ .

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Step 6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.45666638 + 2 π n, 5.82651892 + 2 π n$ , for any integer $n$