Trigonometry Examples

$r = 2 (\sin (x) - \cos (x))$

Step 1

Rewrite the equation as $2 (\sin (x) - \cos (x)) = r$ .

$2 (\sin (x) - \cos (x)) = r$

Step 2

Divide each term in the equation by $\cos (x)$ .

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

Step 3

Replace $\cos (x)$ with an equivalent expression in the numerator.

$2 (\sin (x) - \cos (x)) \cdot \sec (x) = \frac{r}{\cos (x)}$

Step 4

Remove parentheses.

$2 (\sin (x) - \cos (x)) \cdot \sec (x) = \frac{r}{\cos (x)}$

Step 5

Apply the distributive property.

$(2 \sin (x) + 2 (- \cos (x))) \cdot \sec (x) = \frac{r}{\cos (x)}$

Step 6

Multiply $- 1$ by $2$ .

$(2 \sin (x) - 2 \cos (x)) \cdot \sec (x) = \frac{r}{\cos (x)}$

Step 7

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

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

Step 8

Apply the distributive property.

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

Step 9

Multiply $2 \sin (x) \frac{1}{\cos (x)}$ .

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Step 9.1

Combine $\frac{1}{\cos (x)}$ and $2$ .

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

Step 9.2

Combine $\frac{2}{\cos (x)}$ and $\sin (x)$ .

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

Step 10

Cancel the common factor of $\cos (x)$ .

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Step 10.1

Factor $\cos (x)$ out of $- 2 \cos (x)$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

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

Step 11

Simplify each term.

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Step 11.1

Separate fractions.

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

Step 11.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{2}{1} \cdot \tan (x) - 2 = \frac{r}{\cos (x)}$

Step 11.3

Divide $2$ by $1$ .

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

Step 12

Separate fractions.

$2 \tan (x) - 2 = \frac{r}{1} \cdot \frac{1}{\cos (x)}$

Step 13

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

$2 \tan (x) - 2 = \frac{r}{1} \cdot \sec (x)$

Step 14

Divide $r$ by $1$ .

$2 \tan (x) - 2 = r \sec (x)$