Find the Exact Value tan(pi/3)+sin((5pi)/6)*cos(-(3pi)/4)

$\tan (\frac{π}{3}) + \sin (\frac{5 π}{6}) \cdot \cos (- \frac{3 π}{4})$

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

$\sqrt{3} + \sin (\frac{5 π}{6}) \cdot \cos (- \frac{3 π}{4})$

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

$\sqrt{3} + \sin (\frac{π}{6}) \cdot \cos (- \frac{3 π}{4})$

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

$\sqrt{3} + \frac{1}{2} \cdot \cos (- \frac{3 π}{4})$

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

$\sqrt{3} + \frac{1}{2} \cdot \cos (\frac{5 π}{4})$

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.

$\sqrt{3} + \frac{1}{2} \cdot (- \cos (\frac{π}{4}))$

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\sqrt{3} + \frac{1}{2} \cdot (- \frac{\sqrt{2}}{2})$

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

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

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

Multiply $2$ by $2$ .

$\sqrt{3} - \frac{\sqrt{2}}{4}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3} - \frac{\sqrt{2}}{4}$

Decimal Form:

$1.37849741 \dots$

Find the Exact Value tan(pi/3)+sin((5pi)/6)*cos(-(3pi)/4)

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