Trigonometry Examples

$\sin (x) \cos (x) = \sqrt{\frac{3}{4}}$

Step 1

Multiply each term in $\sin (x) \cos (x) = \sqrt{\frac{3}{4}}$ by $2$ to eliminate the fractions.

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Step 1.1

Multiply each term in $\sin (x) \cos (x) = \sqrt{\frac{3}{4}}$ by $2$ .

$\sin (x) \cos (x) \cdot 2 = \sqrt{\frac{3}{4}} \cdot 2$

Step 1.2

Simplify the left side.

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Step 1.2.1

Reorder $\sin (x) \cos (x)$ and $2$ .

$2 \cdot (\sin (x) \cos (x)) = \sqrt{\frac{3}{4}} \cdot 2$

Step 1.2.2

Apply the sine double-angle identity.

$\sin (2 x) = \sqrt{\frac{3}{4}} \cdot 2$

Step 1.3

Simplify the right side.

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Step 1.3.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\sin (2 x) = \frac{\sqrt{3}}{\sqrt{4}} \cdot 2$

Step 1.3.2

Simplify the denominator.

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Step 1.3.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.3.2.2

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

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

Step 1.3.3

Cancel the common factor of $2$ .

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Step 1.3.3.1

Cancel the common factor.

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

Step 1.3.3.2

Rewrite the expression.

$\sin (2 x) = \sqrt{3}$

Step 2

The range of sine is $- 1 \leq y \leq 1$ . Since $\sqrt{3}$ does not fall in this range, there is no solution.

No solution