Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin 105^{\circ}$$

Step-by-Step Solution

Answer

\(\sin 105^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4}\).

Step 1: Identify Relevant Trigonometric Identity

To find \( \sin 105^{\circ} \), we need to express \(105^{\circ}\) as a sum or difference of angles whose sine values are known. We can use the identity for the sine of a sum: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Here, we can write \(105^{\circ}\) as \(60^{\circ} + 45^{\circ}\).

Step 2: Set Up the Identity

Using the identity \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), substitute \( a = 60^{\circ} \) and \( b = 45^{\circ} \). The equation becomes: \[\sin 105^{\circ} = \sin(60^{\circ} + 45^{\circ}) = \sin 60^{\circ} \cos 45^{\circ} + \cos 60^{\circ} \sin 45^{\circ}\]

Step 3: Substitute Known Values

We know the exact trigonometric values: \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), \( \cos 60^{\circ} = \frac{1}{2} \), \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \), and \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \). Substitute these into the equation from Step 2: \[\sin 105^{\circ} = \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{2}}{2} \right) + \left( \frac{1}{2} \right) \left( \frac{\sqrt{2}}{2} \right)\]

Step 4: Simplify the Expression

Calculate the products and then add the terms:the first term: \( \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} \)the second term: \( \frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} \)Thus, \[\sin 105^{\circ} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}\]

Step 5: Present the Final Answer

Now, the exact value of \( \sin 105^{\circ} \) is \[\frac{\sqrt{6} + \sqrt{2}}{4}\].

Key Concepts

Sine of a Sum

The sine of a sum is a useful trigonometric identity that helps us find the sine of an angle expressed as the sum of two other angles. This is especially helpful when dealing with angles that might not have straightforward sine values. The identity is expressed as:\[\sin(a + b) = \sin a \cos b + \cos a \sin b\]Here's how it works:

• Express the angle you're interested in as a sum of two known angles. For example, to find \( \sin 105^{\circ} \), we can rewrite it as \( 60^{\circ} + 45^{\circ}\).
• Apply the identity by substituting the known values of the sine and cosine for the angles.

This approach allows you to break down complex problems into simpler parts and use known values to find precise answers.

Exact Trigonometric Values

Exact trigonometric values are critical when using identities like the sine of a sum. These are values for specific angles that we can easily remember and use. Here are some key ones used in trigonometry:

• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)

Using these exact values, we can calculate angles that are sums or differences of these known angles. For \( \sin 105^{\circ} \), using these precise values ensures an accurate result without relying on a calculator.

Angle Addition Formula

The angle addition formula is a cornerstone in trigonometry, allowing us to find the sine, cosine, or tangent of an angle created by adding or subtracting two known angles. For example, the sine angle addition formula is used like this:\[\sin(a + b) = \sin a \cos b + \cos a \sin b\]In the case of \( \sin 105^{\circ} \), rewriting it as the sum of \( 60^{\circ} \) and \( 45^{\circ} \) is the first step. This is because we know the trigonometric values for 60 and 45 degrees. Understanding this concept helps break complex calculations into manageable steps. By applying these formulas, we transform the task into something solvable using simple arithmetic and known angle values.