Trigonometry Examples

$\cos (3 x) = \sqrt{\frac{3}{2}}$

Step 1

Simplify $\sqrt{\frac{3}{2}}$ .

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Step 1.1

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$\cos (3 x) = \frac{\sqrt{3}}{\sqrt{2}}$

Step 1.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (3 x) = \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.3.2

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

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.3.3

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

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.3.4

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

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.3.5

Add $1$ and $1$ .

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.3.6

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

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Step 1.3.6.1

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

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.3.6.2

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

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.3.6.3

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

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.3.6.4.2

Rewrite the expression.

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{2^{1}}$

Step 1.3.6.5

Evaluate the exponent.

$\cos (3 x) = \frac{\sqrt{3} \sqrt{2}}{2}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\cos (3 x) = \frac{\sqrt{3 \cdot 2}}{2}$

Step 1.4.2

Multiply $3$ by $2$ .

$\cos (3 x) = \frac{\sqrt{6}}{2}$

Step 2

The range of cosine is $- 1 \leq y \leq 1$ . Since $\frac{\sqrt{6}}{2}$ does not fall in this range, there is no solution.

No solution