Problem 127 - Calculus Volume 1

Question

Evaluate the following functional values. $$\sin \left(\frac{\pi}{12}\right)$$

Step-by-Step Solution

Answer

\( \sin \left(\frac{\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4} \).

Step 1: Understanding the Problem

We are tasked with finding the value of \( \sin \left(\frac{\pi}{12}\right) \). This angle is not commonly found in basic trigonometric tables, so we will use angle sum identities to find its value.

Step 2: Choose Appropriate Angle Sum

\( \frac{\pi}{12} \) can be expressed as a sum of angles that we know the sine and cosine values for. Specifically, we note that \( \frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} \). This will allow us to use the identity for the sine of a difference.

Step 3: Apply Sine of a Difference Formula

The formula for the sine of a difference is \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). Applying this, we get: \[sin \left(\frac{\pi}{12}\right) = \sin \left(\frac{\pi}{3} - \frac{\pi}{4}\right)\]\[= \sin \left(\frac{\pi}{3}\right) \cos \left(\frac{\pi}{4}\right) - \cos \left(\frac{\pi}{3}\right) \sin \left(\frac{\pi}{4}\right)\]

Step 4: Substitute Known Values

Now, use the known trigonometric values: - \( \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)- \( \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)- \( \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \)- \( \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)Substitute these into the formula:\[= \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right)\]

Step 5: Simplify the Expression

Multiply these fractions:\[= \frac{\sqrt{3} \times \sqrt{2}}{4} - \frac{1 \times \sqrt{2}}{4}\]\[= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\]Combine like terms:\[= \frac{\sqrt{6} - \sqrt{2}}{4}\]

Step 6: Write Final Answer

The value of \( \sin \left(\frac{\pi}{12}\right) \) is \( \frac{\sqrt{6} - \sqrt{2}}{4} \).

Key Concepts

Angle Sum Identities

The Angle Sum Identities are fundamental tools in trigonometry that allow you to find the sine, cosine, or tangent of a sum or difference of angles. These identities are helpful for calculating the trigonometric values of angles that aren't easily found on the unit circle.
One of the primary identities used for sine is:

• \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \)

To use this identity, you start by expressing the given angle as a sum or difference of angles with known trigonometric values. This transforms a complex problem into a simpler one.
You can then substitute the known values into the identity to calculate the desired sine. Understanding how to split angles efficiently is crucial, making this process a handy technique for evaluating trigonometric functions.

Sine Function

The sine function is a core concept in trigonometry, relating an angle to the ratio of the length of the opposite side to the hypotenuse in a right triangle. In the unit circle framework,

• \( \sin(\theta) \) corresponds to the y-coordinate of a point on the circle.

The sine function is periodic with a period of \( 2\pi \), which means it repeats its values every \( 2\pi \) radians. This function is also symmetric about the origin, making it an odd function, which can be expressed as:

• \( \sin(-\theta) = -\sin(\theta) \)

Its range is from -1 to 1, making it a bounded function. In the exercise, to find \( \sin\left(\frac{\pi}{12}\right) \), we utilized an angle difference identity to break it into components, each with a calculable sine.

Radians

Radians are a way to measure angles based on the radius of a circle. One full rotation around a circle is \( 2\pi \) radians, equivalent to 360 degrees.
Thus, when you divide the circle into smaller parts, each piece's angle can be expressed in radians.
For instance:

• \( \frac{\pi}{4} \) radians is a 45-degree angle.
• \( \frac{\pi}{3} \) radians is a 60-degree angle.

Using radians simplifies many mathematical expressions, particularly in calculus and trigonometry. The exercise asked for \( \sin\left(\frac{\pi}{12}\right) \), which is manageable with radians and angle sum identities, even though \( \frac{\pi}{12} \) isn't one of the commonly memorized angles.

Trigonometric Values

Trigonometric values refer to the known sine, cosine, and tangent values for standard angles that are commonly found on the unit circle, where the circle has a radius of one. These angles typically include:

• 0, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), and their corresponding multiples.

Knowing these values by heart allows you to perform quick calculations and simplifies the application of trigonometric identities.
For example:

• \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)
• \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)

In the exercise, these memorized values were crucial in applying the formula for the sine of a difference to find \( \sin\left(\frac{\pi}{12}\right) \). Understanding and using these known values efficiently can ease the solving of complex trigonometric equations.