Trigonometry Examples

$\sin (\frac{π}{6}) + \cos (\frac{π}{3}) + \sin (\frac{5 π}{6}) + \cos (\frac{17 π}{6})$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{2} + \cos (\frac{π}{3}) + \sin (\frac{5 π}{6}) + \cos (\frac{17 π}{6})$

Step 1.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{1}{2} + \frac{1}{2} + \sin (\frac{5 π}{6}) + \cos (\frac{17 π}{6})$

Step 1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{1}{2} + \frac{1}{2} + \sin (\frac{π}{6}) + \cos (\frac{17 π}{6})$

Step 1.4

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

$\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cos (\frac{17 π}{6})$

Step 1.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cos (\frac{5 π}{6})$

Step 1.6

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 second quadrant.

$\frac{1}{2} + \frac{1}{2} + \frac{1}{2} - \cos (\frac{π}{6})$

Step 1.7

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

$\frac{1}{2} + \frac{1}{2} + \frac{1}{2} - \frac{\sqrt{3}}{2}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{1 + 1 + 1 - \sqrt{3}}{2}$

Step 2.2

Simplify by adding numbers.

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Step 2.2.1

Add $1$ and $1$ .

$\frac{2 + 1 - \sqrt{3}}{2}$

Step 2.2.2

Add $2$ and $1$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.63397459 \dots$