Trigonometry Examples

$\cos (x) = \frac{- 41}{\sqrt{145} \sqrt{106}}$

Step 1

Simplify $\frac{- 41}{\sqrt{145} \sqrt{106}}$ .

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Simplify the denominator.

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Combine using the product rule for radicals.

$\cos (x) = \frac{- 41}{\sqrt{145 \cdot 106}}$

Multiply $145$ by $106$ .

$\cos (x) = \frac{- 41}{\sqrt{15370}}$

Move the negative in front of the fraction.

$\cos (x) = - \frac{41}{\sqrt{15370}}$

Multiply $\frac{41}{\sqrt{15370}}$ by $\frac{\sqrt{15370}}{\sqrt{15370}}$ .

$\cos (x) = - (\frac{41}{\sqrt{15370}} \cdot \frac{\sqrt{15370}}{\sqrt{15370}})$

Combine and simplify the denominator.

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Multiply $\frac{41}{\sqrt{15370}}$ by $\frac{\sqrt{15370}}{\sqrt{15370}}$ .

$\cos (x) = - \frac{41 \sqrt{15370}}{\sqrt{15370} \sqrt{15370}}$

Raise $\sqrt{15370}$ to the power of $1$ .

$\cos (x) = - \frac{41 \sqrt{15370}}{{\sqrt{15370}}^{1} \sqrt{15370}}$

Raise $\sqrt{15370}$ to the power of $1$ .

$\cos (x) = - \frac{41 \sqrt{15370}}{{\sqrt{15370}}^{1} {\sqrt{15370}}^{1}}$

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

$\cos (x) = - \frac{41 \sqrt{15370}}{{\sqrt{15370}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos (x) = - \frac{41 \sqrt{15370}}{{\sqrt{15370}}^{2}}$

Rewrite ${\sqrt{15370}}^{2}$ as $15370$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15370}$ as $15370^{\frac{1}{2}}$ .

$\cos (x) = - \frac{41 \sqrt{15370}}{{(15370^{\frac{1}{2}})}^{2}}$

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

$\cos (x) = - \frac{41 \sqrt{15370}}{15370^{\frac{1}{2} \cdot 2}}$

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

$\cos (x) = - \frac{41 \sqrt{15370}}{15370^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\cos (x) = - \frac{41 \sqrt{15370}}{15370^{\frac{2}{2}}}$

Rewrite the expression.

$\cos (x) = - \frac{41 \sqrt{15370}}{15370^{1}}$

Evaluate the exponent.

$\cos (x) = - \frac{41 \sqrt{15370}}{15370}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 1.9078518 + 2 π n, 4.37533349 + 2 π n$ , for any integer $n$

Step 3