Problem 46 - Thomas Calculus

Question

Evaluate sin \(\frac{5 \pi}{12}\).

Step-by-Step Solution

Answer

The value of \(\sin \frac{5\pi}{12}\) is \(\frac{\sqrt{6} + \sqrt{2}}{4}\).

Step 1: Identify Known Angles

The angle \(\frac{5\pi}{12}\) is not a standard angle with a known sine value. We need to express it as a sum or difference of angles for which the sine values are known. We choose \(\frac{5\pi}{12}\) as the sum of \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\).

Step 2: Use the Sine Identity for Sum of Angles

The sine of the sum of two angles \(A\) and \(B\) is given by \(\sin(A+B) = \sin A \cos B + \cos A \sin B\). Here, choose \(A = \frac{\pi}{4}\) and \(B = \frac{\pi}{6}\).

Step 3: Calculate Necessary Trigonometric Values

Calculate the sine and cosine values for \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\):- \(\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\)- \(\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}\)- \(\sin\frac{\pi}{6} = \frac{1}{2}\)- \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\)

Step 4: Substitute and Simplify

Substitute the calculated trigonometric values into \(\sin(A+B)\):\[\sin\left(\frac{5\pi}{12}\right) = \sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right).\]Simplify the expression:\[= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}.\]

Step 5: Write the Final Answer

The evaluated value of \(\sin \frac{5\pi}{12}\) is \(\frac{\sqrt{6} + \sqrt{2}}{4}\).

Key Concepts

Sum of Angles Identity

The sum of angles identity is a fundamental concept in trigonometry. It allows us to find the sine, cosine, or tangent of an angle expressed as the sum or difference of two known angles. For this exercise, we're focusing on understanding how to use this identity to evaluate the sine of a sum of angles.
The identity for the sine of a sum of two angles, say \(A\) and \(B\), is:

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

This identity breaks down the sine of a more complex angle into a combination of simpler trigonometric functions. It can be particularly useful when dealing with angles like \(\frac{5\pi}{12}\), which aren't standard angles with known sine values. By expressing such angles as a sum (or difference) of angles with known values, we can evaluate trigonometric functions more easily.

Sine Function Evaluation

Understanding the sine function and its properties is essential to evaluate it accurately for different angles. The sine function, part of the trigonometric functions family, measures the y-coordinate or vertical component of a point on the unit circle.
If we need to determine \(\sin \frac{5\pi}{12}\), we realize first that it's not one of the common angles like \(\pi/6\), \(\pi/4\), or \(\pi/3\). Therefore, we use the sum of angles approach. By expressing \(\frac{5\pi}{12}\) as \(\frac{\pi}{4} + \frac{\pi}{6}\), we can use their known sine and cosine values:

• \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)
• \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)
• \(\sin \frac{\pi}{6} = \frac{1}{2}\)
• \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)

These known values are then applied in the identity formula for the sine of a sum to solve for the desired angle. This process allows us to leverage established trigonometric values to find the sine of more complex angles.

Angle Sum Formula

The angle sum formula is closely related to the sum of angles identity but extends it by providing a composite formula for all the main trigonometric functions when applied to sums or differences of angles. For example, when dealing with angles not commonly found on the unit circle, this formula becomes indispensable.
In trigonometry, the angle sum formulas are as follows:

• For sine: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
• For cosine: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
• For tangent: \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)

These formulas allow for the evaluation of trigonometric functions without directly computing difficult angles. In this specific exercise, the sine formula is directly applied to solve \(\sin \frac{5\pi}{12}\), but understanding that this approach can be generalized to cosine and tangent as well enriches our toolkit for handling a variety of trigonometric problems.