Calculus Examples

$f (\frac{7 π}{6}) = 3 \sin (x) + 9 \cos (x)$

Step 1

Replace the variable $x$ with $\frac{7 π}{6}$ in the expression.

$f (\frac{7 π}{6}) = 3 \sin (\frac{7 π}{6}) + 9 \cos (\frac{7 π}{6})$

Step 2

Simplify the result.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$f (\frac{7 π}{6}) = 3 (- \sin (\frac{π}{6})) + 9 \cos (\frac{7 π}{6})$

Step 2.1.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$f (\frac{7 π}{6}) = 3 (- \frac{1}{2}) + 9 \cos (\frac{7 π}{6})$

Step 2.1.3

Multiply $3 (- \frac{1}{2})$ .

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Step 2.1.3.1

Multiply $- 1$ by $3$ .

$f (\frac{7 π}{6}) = - 3 (\frac{1}{2}) + 9 \cos (\frac{7 π}{6})$

Step 2.1.3.2

Combine $- 3$ and $\frac{1}{2}$ .

$f (\frac{7 π}{6}) = \frac{- 3}{2} + 9 \cos (\frac{7 π}{6})$

Step 2.1.4

Move the negative in front of the fraction.

$f (\frac{7 π}{6}) = - \frac{3}{2} + 9 \cos (\frac{7 π}{6})$

Step 2.1.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$f (\frac{7 π}{6}) = - \frac{3}{2} + 9 (- \cos (\frac{π}{6}))$

Step 2.1.6

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$f (\frac{7 π}{6}) = - \frac{3}{2} + 9 (- \frac{\sqrt{3}}{2})$

Step 2.1.7

Multiply $9 (- \frac{\sqrt{3}}{2})$ .

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Step 2.1.7.1

Multiply $- 1$ by $9$ .

$f (\frac{7 π}{6}) = - \frac{3}{2} - 9 \frac{\sqrt{3}}{2}$

Step 2.1.7.2

Combine $- 9$ and $\frac{\sqrt{3}}{2}$ .

$f (\frac{7 π}{6}) = - \frac{3}{2} + \frac{- 9 \sqrt{3}}{2}$

Step 2.1.8

Move the negative in front of the fraction.

$f (\frac{7 π}{6}) = - \frac{3}{2} - \frac{9 \sqrt{3}}{2}$

Step 2.2

The final answer is $- \frac{3}{2} - \frac{9 \sqrt{3}}{2}$ .

$- \frac{3}{2} - \frac{9 \sqrt{3}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{2} - \frac{9 \sqrt{3}}{2}$

Decimal Form:

$- 9.29422863 \dots$

Step 4