Trigonometry Examples

$\sin (x) \cot (x) + \frac{1}{\sqrt{2}} = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1.1

Reorder $\sin (x)$ and $\cot (x)$ .

$\cot (x) \sin (x) + \frac{1}{\sqrt{2}} = 0$

Step 1.1.1.2

Rewrite $\sin (x) \cot (x)$ in terms of sines and cosines.

$\frac{\cos (x)}{\sin (x)} \sin (x) + \frac{1}{\sqrt{2}} = 0$

Step 1.1.1.3

Cancel the common factors.

$\cos (x) + \frac{1}{\sqrt{2}} = 0$

Step 1.1.2

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

$\cos (x) + \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = 0$

Step 1.1.3

Combine and simplify the denominator.

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Step 1.1.3.1

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

$\cos (x) + \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}} = 0$

Step 1.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) + \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} = 0$

Step 1.1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) + \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} = 0$

Step 1.1.3.4

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

$\cos (x) + \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}} = 0$

Step 1.1.3.5

Add $1$ and $1$ .

$\cos (x) + \frac{\sqrt{2}}{{\sqrt{2}}^{2}} = 0$

Step 1.1.3.6

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

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Step 1.1.3.6.1

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

$\cos (x) + \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} = 0$

Step 1.1.3.6.2

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

$\cos (x) + \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}} = 0$

Step 1.1.3.6.3

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

$\cos (x) + \frac{\sqrt{2}}{2^{\frac{2}{2}}} = 0$

Step 1.1.3.6.4

Cancel the common factor of $2$ .

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Step 1.1.3.6.4.1

Cancel the common factor.

$\cos (x) + \frac{\sqrt{2}}{2^{\frac{2}{2}}} = 0$

Step 1.1.3.6.4.2

Rewrite the expression.

$\cos (x) + \frac{\sqrt{2}}{2^{1}} = 0$

Step 1.1.3.6.5

Evaluate the exponent.

$\cos (x) + \frac{\sqrt{2}}{2} = 0$

Step 2

Subtract $\frac{\sqrt{2}}{2}$ from both sides of the equation.

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 5

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

$x = 2 π - \frac{3 π}{4}$

Step 6

Simplify $2 π - \frac{3 π}{4}$ .

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Step 6.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 6.2

Combine fractions.

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Step 6.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 6.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$