Trigonometry Examples

$r (5 \cos (x) + 6 \sin (x)) = 11$

Step 1

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

$\frac{r (5 \cos (x) + 6 \sin (x))}{\cos (x)} = \frac{11}{\cos (x)}$

Step 2

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

$r (5 \cos (x) + 6 \sin (x)) \cdot \sec (x) = \frac{11}{\cos (x)}$

Step 3

Remove parentheses.

$r (5 \cos (x) + 6 \sin (x)) \cdot \sec (x) = \frac{11}{\cos (x)}$

Step 4

Apply the distributive property.

$(r (5 \cos (x)) + r (6 \sin (x))) \cdot \sec (x) = \frac{11}{\cos (x)}$

Step 5

Reorder.

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Step 5.1

Rewrite using the commutative property of multiplication.

$(5 r \cos (x) + r (6 \sin (x))) \cdot \sec (x) = \frac{11}{\cos (x)}$

Step 5.2

Rewrite using the commutative property of multiplication.

$(5 r \cos (x) + 6 r \sin (x)) \cdot \sec (x) = \frac{11}{\cos (x)}$

Step 6

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

$(5 r \cos (x) + 6 r \sin (x)) \cdot \frac{1}{\cos (x)} = \frac{11}{\cos (x)}$

Step 7

Apply the distributive property.

$5 r \cos (x) (\frac{1}{\cos (x)}) + 6 r \sin (x) (\frac{1}{\cos (x)}) = \frac{11}{\cos (x)}$

Step 8

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

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Step 8.1

Factor $\cos (x)$ out of $5 r \cos (x)$ .

$\cos (x) (5 r) (\frac{1}{\cos (x)}) + 6 r \sin (x) (\frac{1}{\cos (x)}) = \frac{11}{\cos (x)}$

Step 8.2

Cancel the common factor.

$\cos (x) (5 r) (\frac{1}{\cos (x)}) + 6 r \sin (x) (\frac{1}{\cos (x)}) = \frac{11}{\cos (x)}$

Step 8.3

Rewrite the expression.

$5 r + 6 r \sin (x) (\frac{1}{\cos (x)}) = \frac{11}{\cos (x)}$

Step 9

Multiply $6 r \sin (x) \frac{1}{\cos (x)}$ .

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Step 9.1

Combine $6$ and $\frac{1}{\cos (x)}$ .

$5 r + r \sin (x) (\frac{6}{\cos (x)}) = \frac{11}{\cos (x)}$

Step 9.2

Combine $\frac{6}{\cos (x)}$ and $r$ .

$5 r + \frac{6 r}{\cos (x)} \cdot \sin (x) = \frac{11}{\cos (x)}$

Step 9.3

Combine $\frac{6 r}{\cos (x)}$ and $\sin (x)$ .

$5 r + \frac{6 r \sin (x)}{\cos (x)} = \frac{11}{\cos (x)}$

Step 10

Simplify each term.

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Step 10.1

Separate fractions.

$5 r + \frac{6 r}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{11}{\cos (x)}$

Step 10.2

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

$5 r + \frac{6 r}{1} \cdot \tan (x) = \frac{11}{\cos (x)}$

Step 10.3

Divide $6 r$ by $1$ .

$5 r + 6 r \tan (x) = \frac{11}{\cos (x)}$

Step 11

Separate fractions.

$5 r + 6 r \tan (x) = \frac{11}{1} \cdot \frac{1}{\cos (x)}$

Step 12

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

$5 r + 6 r \tan (x) = \frac{11}{1} \cdot \sec (x)$

Step 13

Divide $11$ by $1$ .

$5 r + 6 r \tan (x) = 11 \sec (x)$