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

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

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

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

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 second quadrant.

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

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

$\sqrt{3} - \frac{\sqrt{3}}{2} \cdot \sin (- \frac{3 π}{4})$

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

$\sqrt{3} - \frac{\sqrt{3}}{2} \cdot \sin (\frac{5 π}{4})$

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

$\sqrt{3} - \frac{\sqrt{3}}{2} \cdot (- \sin (\frac{π}{4}))$

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

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

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{2}}{2}$ .

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

$\sqrt{3} + \frac{\sqrt{6}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\sqrt{3} + \frac{\sqrt{6}}{4}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3} + \frac{\sqrt{6}}{4}$

Decimal Form:

$2.34442324 \dots$

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

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