Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-10: Find the exact values. (a) \(\sin \frac{2 \pi}{3}+\sin \frac{\pi}{4}\) (b) \(\sin \frac{11 \pi}{12} \quad\left(\right.\) use \(\left.\frac{11 \pi}{12}=\frac{2 \pi}{3}+\frac{\pi}{4}\right)\)

Step-by-Step Solution

Answer

(a) \( \frac{\sqrt{3} + \sqrt{2}}{2} \), (b) \( \frac{\sqrt{6} - \sqrt{2}}{4} \)

Step 1: Find Exact Value of Sine for Each Angle for Part (a)

For part (a), we need to find the exact values of \( \sin \frac{2\pi}{3} \) and \( \sin \frac{\pi}{4} \).1. \( \sin \frac{2\pi}{3} = \sin 120^\circ \). The angle \(120^\circ\) is in the second quadrant where sine is positive. Its reference angle is \(60^\circ\), hence \( \sin \frac{2\pi}{3} = \sin 60^\circ = \frac{\sqrt{3}}{2} \).2. \( \sin \frac{\pi}{4} = \sin 45^\circ = \frac{\sqrt{2}}{2} \).

Step 2: Add Sine Values for Part (a)

Now, add the sine values:\[ \sin \frac{2\pi}{3} + \sin \frac{\pi}{4} = \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \]

Step 3: Use Angle Sum Formula for Part (b)

For part (b), we are given \( \sin \frac{11\pi}{12} \) which can be rewritten as \( \sin\left(\frac{2\pi}{3} + \frac{\pi}{4}\right) \).The angle sum formula for sine is:\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]

Step 4: Compute Values for Angle Sum Formula

Using the angles \(A = \frac{2\pi}{3}\) and \(B = \frac{\pi}{4}\):1. \( \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2},\ \cos \frac{2\pi}{3} = -\frac{1}{2} \) (since cosine is negative in the second quadrant).2. \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2},\ \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).Plug these into the formula:\[ \sin \frac{11\pi}{12} = \sin \frac{2\pi}{3} \cos \frac{\pi}{4} + \cos \frac{2\pi}{3} \sin \frac{\pi}{4} \]\[ = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(-\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) \]

Step 5: Simplify Results for Part (b)

Simplify the expression:\[ \sin \frac{11\pi}{12} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \]Combine the terms:\[ = \frac{\sqrt{6} - \sqrt{2}}{4} \]

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often abbreviated as "sin". It relates the angle of a right triangle to the ratio of the length of the side opposite that angle to the length of the hypotenuse.
The function is periodic, with a period of \(2\pi\), and its values range from -1 to 1.

• In the unit circle, the sine of an angle \(\theta\) is the y-coordinate of the point where the terminal side of the angle intersects the circle.
• The sine function is positive in the first and second quadrants of the unit circle.
• Common angles often used in calculations are 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), and others, including their corresponding angles in other quadrants.

By understanding these properties, you can effectively navigate problems involving \(\sin(\theta)\). This makes it crucial for solving various trigonometric problems, such as the one given in the exercise above.

Angle Sum Formula

The angle sum formula for sine is an important trigonometric identity used to find the sine of the sum of two angles. The formula is given by:
\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]This formula allows you to break down complex expressions into simpler calculations, using known values of sine and cosine for the individual angles.

• For example, if you need to calculate \(\sin \left(\frac{2\pi}{3} + \frac{\pi}{4}\right)\), you substitute the sine and cosine values of \(\frac{2\pi}{3}\) and \(\frac{\pi}{4}\) into the formula.
• The sine and cosine terms are calculated based on their respective angles.

Using this method makes complicated trigonometric problems more approachable by applying simpler calculations from individual trigonometric functions.

Exact Values of Trigonometric Functions

Knowing the exact values of trigonometric functions for specific angles is extremely helpful in solving trigonometric equations. Exact values mean using fractions and square roots, rather than decimal approximations. Common angles like \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) have well-known exact values.

• For instance, \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\). This means the opposite side and hypotenuse of a 45° triangle are proportional to \(\sqrt{2}\) and 2, respectively.
• Also, \(\cos \frac{2\pi}{3} = -\frac{1}{2}\) since this angle is in the second quadrant, where cosine values are negative.

Being familiar with these values enhances your ability to quickly solve trigonometric equations without a calculator, especially when working with standard angles on the unit circle.