Problem 11 - Precalculus with Calculus Previews

Question

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin \frac{11 \pi}{12} $$

Step-by-Step Solution

Answer

\( \sin \frac{11\pi}{12} = \frac{\sqrt{2}}{2} \).

Step 1: Identify the formula

Our task is to find the exact value of \( \sin \frac{11\pi}{12} \). We'll use a sum or difference identity for sine to simplify this expression. We can start by expressing \( \frac{11\pi}{12} \) as the sum of angles whose sine values we know.

Step 2: Express the angle as a sum

We express \( \frac{11\pi}{12} \) as a sum of two angles: \( \frac{11\pi}{12} = \frac{6\pi}{12} + \frac{5\pi}{12} = \frac{\pi}{2} + \frac{\pi}{4} \). This way, we can utilize the sine sum identity.

Step 3: Apply the sine sum identity

We use the sine sum identity: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). Here, \( A = \frac{\pi}{2} \) and \( B = \frac{\pi}{4} \). Plugging in the values: \( \sin \frac{11\pi}{12} = \sin \left(\frac{\pi}{2} + \frac{\pi}{4}\right) = \sin \frac{\pi}{2} \cos \frac{\pi}{4} + \cos \frac{\pi}{2} \sin \frac{\pi}{4} \).

Step 4: Substitute and simplify

Now substitute the known trigonometric values: \( \sin \frac{\pi}{2} = 1 \), \( \cos \frac{\pi}{2} = 0 \), \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), and \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). Substitute these into the identity: \( \sin \frac{11\pi}{12} = 1 \cdot \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \).

Key Concepts

Sine Sum Formula

The sine sum formula is a valuable trigonometric identity that helps in finding the sine of the sum of two angles. This formula is written as \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). It breaks down the sine of a sum into components that are often easier to find individually, using known sine and cosine values. When you have a complex angle, the sine sum formula allows you to express it in terms of simpler, well-known angles. This is particularly useful because the exact values of sine and cosine are known for special angles like \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \text{ and } \frac{\pi}{2} \). Using this formula makes it easier to work with angles that are sums or differences since you're able to convert them into terms that are more manageable. This method simplifies calculations and helps to avoid the need for approximation or using a calculator. Always remember to break down your angle into familiar components and apply the formula correctly to achieve accurate results.

Angle Addition

Angle addition is a primary method used in trigonometry that involves breaking down complex angles into simpler parts. This technique plays a crucial role in the application of trigonometric identities, such as the sine sum formula. For instance, given an angle like \( \frac{11\pi}{12} \), it can be challenging to determine its exact sine value directly. By expressing \( \frac{11\pi}{12} \) as a sum of simpler angles such as \( \frac{\pi}{2} + \frac{\pi}{4} \), you can easily utilize known values of \( \sin \frac{\pi}{2} \) and \( \sin \frac{\pi}{4} \). This simplification makes calculation more straightforward.

• Make sure the angles you choose have known trigonometric values.
• Check the arithmetic thoroughly as errors in addition can lead to wrong identities in computation.
• Apply the relevant trigonometric identity to evaluate the exact value correctly.

Breaking down angles into sums or differences with easily remembered counterparts is a powerful technique for solving trigonometric expressions.

Exact Values of Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent have specific exact values at certain angles. These exact values are defined in terms of simple fractions and square roots, which are the keys to manually solving trigonometric expressions.For example, the exact value of \( \sin \frac{\pi}{2} \) is 1, \( \cos \frac{\pi}{2} \) is 0, \( \sin \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \), and \( \cos \frac{\pi}{4} \) is also \( \frac{\sqrt{2}}{2} \). These values are derived from special properties of a unit circle and symmetry of triangles.

• Memorizing these values can significantly ease the process of solving trigonometric equations without a calculator.
• Understanding the derivation of these exact values through geometrical reasoning reinforces knowledge.

When you have angles that can be decomposed using these known values, you dramatically simplify the process of finding the trigonometric function's exact value. This technique not only saves calculation time but also improves accuracy by avoiding numerical approximations.