Trigonometry Examples

cos (x) = \frac{1}{\sqrt{2}}

$\cos (x) = \frac{1}{\sqrt{2}}$

Step 1

Simplify

\frac{1}{\sqrt{2}}

$\frac{1}{\sqrt{2}}$ .

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Step 1.1

Multiply

\frac{1}{\sqrt{2}}

$\frac{1}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

cos (x) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}

$\cos (x) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply

\frac{1}{\sqrt{2}}

$\frac{1}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

cos (x) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}

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

Step 1.2.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}

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

Step 1.2.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}

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

Step 1.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}

$\cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.5

Add

1

$1$ and

1

$1$ .

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

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

Step 1.2.6

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 1.2.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

cos (x) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}

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

Step 1.2.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

cos (x) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}

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

Step 1.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 1.2.6.4

Cancel the common factor of

$2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Evaluate the exponent.

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

Step 2

Take the inverse cosine of both sides of the equation to extract

$x$ from inside the cosine.

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 3

Simplify the right side.

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Step 3.1

The exact value of

$\arccos (\frac{\sqrt{2}}{2})$ is

$\frac{π}{4}$ .

$x = \frac{π}{4}$

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.

$x = 2 π - \frac{π}{4}$

Step 5

Simplify

$2 π - \frac{π}{4}$ .

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Step 5.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.2

Combine fractions.

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Step 5.2.1

Combine

$2 π$ and

$\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 5.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Multiply

$4$ by

$2$ .

$x = \frac{8 π - π}{4}$

Step 5.3.2

Subtract

$π$ from

$8 π$ .

$x = \frac{7 π}{4}$

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.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer

$n$