Problem 5 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Find the exact value of the expression. \(\sin \frac{23 \pi}{12}\)

Step-by-Step Solution

Answer

The exact value of the expression \(\sin \frac{23 \pi}{12}\) is \(\frac{\sqrt{2} + \sqrt{6}}{4}\).

Step 1: Convert the Angle

Find a combination of the given known values \(\pi/4, \pi/6,\) and \(\pi/3\) such that their total is equal to the given value.\nThe suitable combination is \(\frac{3\pi}{4}\) and \(\frac{pi}{3}\) since \(\frac{3\pi}{4} + \frac{\pi}{3} = \frac{9\pi}{12} + \frac{4\pi}{12} = \frac{23\pi}{12}\), the original expression.

Step 2: Apply the Sum-Difference Identity

Using the trigonometric identity \(\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta\), where \(\alpha = \frac{3\pi}{4}\) and \(\beta = \frac{\pi}{3}\), the expression \(\sin \frac{23 \pi}{12}\) transforms into \(\sin(\frac{3\pi}{4}+\frac{\pi}{3}) = \sin\frac{3\pi}{4}\cos\frac{\pi}{3} + \cos\frac{3\pi}{4}\sin\frac{\pi}{3}\).

Step 3: Calculate the Result

Substitute \(\sin\frac{3\pi}{4}\), \(\cos\frac{\pi}{3}\), \(\cos\frac{3\pi}{4}\), and \(\sin\frac{\pi}{3}\) with their exact values \(\sqrt{2}/2\), \(1/2\), \(\sqrt{2}/2\), and \(\sqrt{3}/2\), respectively. Performing these substitutions gives \((\sqrt{2}/2)*(1/2) + \sqrt{2}/2*\sqrt{3}/2) = \sqrt{2}/4 + \sqrt{6}/4 = (\sqrt{2} + \sqrt{6})/4\).

Key Concepts

Sum-Difference Identity

The sum-difference identity is a handy tool in trigonometry, especially when dealing with the sine and cosine of sums or differences of angles. It allows us to break down more complex angles into simpler components, which we can evaluate more easily. The sum identity for sine is: \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \).In this formula, \( \alpha \) and \( \beta \) represent angles. By using this identity, we can compute the sine of an angle expressed as the sum of two known angles.

• The identity helps to transform an expression like \( \sin \frac{23 \pi}{12} \) into a sum of two known angles \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \).
• Once rewritten using the sum identity, we can find each value accurately based on the known angles.

Understanding the sum-difference identity is essential in trigonometry because it simulates breaking down complex problems into manageable pieces.

Exact Values

Unlike decimal approximations, exact values in trigonometry help in maintaining precision and clarity in calculations. They are typically expressed using radicals and fractions. For example, the exact values of \( \sin \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \), and \( \sin \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{2} \).

• Trigonometric exact values eliminate rounding errors, which often occur when using decimal forms.
• They allow mathematical consistency, especially in problems involving angles derived from special right triangles.
• In our exercise, exact values were vital, giving us the accurate result for \( \sin \frac{23 \pi}{12} \).

It's crucial to familiarize yourself with common exact trigonometric values, like those for angles \( \pi/6, \pi/4, \pi/3, \text{and} \pi/2 \), as they form the backbone of many trigonometric calculations.

Angle Conversion

Angle conversion can be an essential step in solving trigonometry problems, especially when you're working with complex angles such as \( \frac{23\pi}{12} \). The idea is to express these angles in terms of more familiar angles, often those that match well with known trigonometric values.

• This process involves converting larger angles into sums or differences of smaller, standard angles, like our example of \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \).
• Knowing which angles can add up to your target angle is the first step in simplification.
• Alternatively, you may use the identity \( \alpha + \beta = \text{target angle} \) to help identify potential known values to use.

Making these conversions requires understanding the circular nature of the trigonometric system, particularly in radians, which allows you to find precise solutions and effectively apply trigonometric identities. This process is crucial for applying identities such as the sum-difference identity, which depends on rephrasing complex angles.