Problem 2 - Algebra 2

Question

Find the exact value of each expression. \(\sin 165^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \( \sin 165^{\circ} \) is \( \frac{\sqrt{6} - \sqrt{2}}{4} \).

Step 1: Identify the Formula to Use

To find the exact value of \( \sin 165^{\circ} \), we recognize that it can be expressed using the sine addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). In this case, we let \( A = 120^{\circ} \) and \( B = 45^{\circ} \). Thus, \( \sin 165^{\circ} = \sin(120^{\circ} + 45^{\circ}) \).

Step 2: Apply the Sine Addition Formula

Using the sine addition formula from the previous step, we get:\[\sin 165^{\circ} = \sin(120^{\circ} + 45^{\circ}) = \sin 120^{\circ} \cos 45^{\circ} + \cos 120^{\circ} \sin 45^{\circ}\]

Step 3: Evaluate Each Trigonometric Function

Now, we need to evaluate each of the trigonometric functions:- \( \sin 120^{\circ} = \sin(180^{\circ} - 60^{\circ}) = \sin 60^{\circ} = \frac{\sqrt{3}}{2} \).- \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).- \( \cos 120^{\circ} = \cos(180^{\circ} - 60^{\circ}) = -\cos 60^{\circ} = -\frac{1}{2} \).- \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

Step 4: Substitute Values and Simplify

Substitute the values found in Step 3 into the expression from Step 2:\[\sin 165^{\circ} = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) + \left(-\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right)\]Simplifying each term:\[\frac{\sqrt{3} \times \sqrt{2}}{4} + \frac{-1 \times \sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}\]Hence, \( \sin 165^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions like sine, cosine, and tangent that are true for any angle. These identities help in simplifying expressions and solving trigonometric equations. One of the most fundamental identities is the sine addition formula:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

This formula is used to find the sine of the sum of two angles. In the given exercise, this identity helps us express \( \sin 165^{\circ} \) as a combination of the sine and cosine of simpler angles. Using identities means recognizing patterns and applying these known equations to find exact values instead of relying on a calculator.

Angle Addition

Angle addition is a technique where a given angle is expressed as the sum or difference of two known angles, whose trigonometric values are easier to evaluate. In the exercise, \( 165^{\circ} \) is decomposed into \( 120^{\circ} + 45^{\circ} \). Both of these angles are special angles with known exact trigonometric values. By breaking down \( 165^{\circ} \), it becomes easier to find the exact value using the sine addition formula. Consider:

• \( 120^{\circ} \) is a known angle from the unit circle, related to \( 60^{\circ} \).
• \( 45^{\circ} \) is another known angle that usually has related symmetry in its trigonometric values.

This shows the importance of choosing angles which simplify derivations, making the solution process straightforward and the results exact.

Exact Trigonometric Values

Exact trigonometric values are the precise outcomes of trigonometric functions for specific angles, often expressed using radicals. These values are not approximated and are crucial when precise answers are needed, as seen in this exercise. Here are some common exact values:

• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 120^{\circ} = -\frac{1}{2} \)

Using these known values, expressions like \( \sin 165^{\circ} \) can be broken down into easily calculated terms. This approach highlights the power of combining exact values from the unit circle with identity equations to solve complex problems without a calculator.