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

Solution:

Step 1:

Compute the exact value of $\tan\left(\frac{\pi}{3}\right)$, which is $\sqrt{3}$. The expression becomes $\sqrt{3} + \cos\left(\frac{5\pi}{6}\right) \cdot \sin\left(-\frac{3\pi}{4}\right)$.

Step 2:

For $\cos\left(\frac{5\pi}{6}\right)$, use its reference angle $\frac{\pi}{6}$, knowing that cosine is negative in the second quadrant. The expression is now $\sqrt{3} - \cos\left(\frac{\pi}{6}\right) \cdot \sin\left(-\frac{3\pi}{4}\right)$.

Step 3:

The exact value of $\cos\left(\frac{\pi}{6}\right)$ is $\frac{\sqrt{3}}{2}$. Update the expression to $\sqrt{3} - \frac{\sqrt{3}}{2} \cdot \sin\left(-\frac{3\pi}{4}\right)$.

Step 4:

To find the equivalent positive angle for $\sin\left(-\frac{3\pi}{4}\right)$, add $2\pi$ to get $\sin\left(\frac{5\pi}{4}\right)$. The expression is now $\sqrt{3} - \frac{\sqrt{3}}{2} \cdot \sin\left(\frac{5\pi}{4}\right)$.

Step 5:

For $\sin\left(\frac{5\pi}{4}\right)$, use its reference angle $\frac{\pi}{4}$, knowing that sine is negative in the third quadrant. The expression becomes $\sqrt{3} - \frac{\sqrt{3}}{2} \cdot \left(-\sin\left(\frac{\pi}{4}\right)\right)$.

Step 6:

The exact value of $\sin\left(\frac{\pi}{4}\right)$ is $\frac{\sqrt{2}}{2}$. Substitute this into the expression to get $\sqrt{3} - \frac{\sqrt{3}}{2} \cdot \left(-\frac{\sqrt{2}}{2}\right)$.

Step 7:

Perform the multiplication of $- \frac{\sqrt{3}}{2} \cdot \left(-\frac{\sqrt{2}}{2}\right)$.

Step 7.1:

Multiply $-1$ by $-1$ to get $1$. The expression is now $\sqrt{3} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$.

Step 7.2:

Multiply $\frac{\sqrt{3}}{2}$ by $1$ to maintain its value. The expression remains $\sqrt{3} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$.

Step 7.3:

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{2}}{2}$ to get $\frac{\sqrt{3} \cdot \sqrt{2}}{4}$.

Step 7.4:

Combine the radicals using the product rule to get $\frac{\sqrt{3 \cdot 2}}{4}$.

Step 7.5:

Simplify the radical to get $\frac{\sqrt{6}}{4}$. The expression is now $\sqrt{3} + \frac{\sqrt{6}}{4}$.

Step 7.6:

The final expression is $\sqrt{3} + \frac{\sqrt{6}}{4}$.

Step 8:

The result can be expressed in different forms. The exact form is $\sqrt{3} + \frac{\sqrt{6}}{4}$, and the decimal form is approximately $2.34442324 \ldots$.

Knowledge Notes:

1. Trigonometric Functions: Trigonometric functions like sine, cosine, and tangent have specific values for standard angles such as $\frac{\pi}{3}$, $\frac{\pi}{4}$, and $\frac{\pi}{6}$.

2. Reference Angles: A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It is used to find the trigonometric function values for angles in different quadrants.

3. Quadrants and Signs: In the unit circle, the sign of the sine and cosine functions depends on the quadrant in which the angle lies. Cosine is negative in the second quadrant, and sine is negative in the third and fourth quadrants.

4. Radians: Radians are a unit of angular measure used in many areas of mathematics. One full rotation (360 degrees) is equal to $2\pi$ radians.

5. Product Rule for Radicals: The product rule for radicals states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, which simplifies the multiplication of square roots.

6. Exact Values: Some trigonometric expressions can be simplified to exact values without using a calculator, which is particularly useful in precise calculations and proofs.