Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\frac{\sin (3 x)}{\sin (x)}$ as a product.

$\sin (3 x) \frac{1}{\sin (x)} - \frac{\cos (3 x)}{\cos (x)} = 2$

Step 1.2

Write $\sin (3 x)$ as a fraction with denominator $1$ .

$\frac{\sin (3 x)}{1} \cdot \frac{1}{\sin (x)} - \frac{\cos (3 x)}{\cos (x)} = 2$

Step 1.3

Simplify.

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Step 1.3.1

Divide $\sin (3 x)$ by $1$ .

$\sin (3 x) \frac{1}{\sin (x)} - \frac{\cos (3 x)}{\cos (x)} = 2$

Step 1.3.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\sin (3 x) \csc (x) - \frac{\cos (3 x)}{\cos (x)} = 2$

Step 1.4

Rewrite $\frac{\cos (3 x)}{\cos (x)}$ as a product.

$\sin (3 x) \csc (x) - (\cos (3 x) \frac{1}{\cos (x)}) = 2$

Step 1.5

Write $\cos (3 x)$ as a fraction with denominator $1$ .

$\sin (3 x) \csc (x) - (\frac{\cos (3 x)}{1} \cdot \frac{1}{\cos (x)}) = 2$

Step 1.6

Simplify.

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Step 1.6.1

Divide $\cos (3 x)$ by $1$ .

$\sin (3 x) \csc (x) - (\cos (3 x) \frac{1}{\cos (x)}) = 2$

Step 1.6.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sin (3 x) \csc (x) - (\cos (3 x) \sec (x)) = 2$

$\sin (3 x) \csc (x) - \cos (3 x) \sec (x) = 2$

Step 2

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

$x \approx 0, - 0.1, 0.1, - \frac{1}{5}, \frac{1}{5}, - 0.3, 0.3$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x \approx 0, - 0.1, 0.1, - \frac{1}{5}, \frac{1}{5}, - 0.3, 0.3$

Decimal Form:

$x \approx 0, - 0.1, 0.1, - 0.2, 0.2, - 0.3, 0.3$

Step 4