Problem 9 - Precalculus: Functions and Graphs

Question

Exer. \(5-10:\) Find the exact values. (a) \(\sin \frac{3 \pi}{4}-\sin \frac{\pi}{6}\) b) \(\sin \frac{7 \pi}{12}\left(\text { use } \frac{7 \pi}{12}=\frac{3 \pi}{4}-\frac{\pi}{6}\right)\)

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{2} - 1}{2}\) (b) \(\frac{\sqrt{6} - \sqrt{2}}{4}\)

Step 1: Calculate \(\sin \frac{3\pi}{4}\)

The angle \(\frac{3\pi}{4}\) is in the second quadrant where sine is positive. The reference angle is \(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\). Therefore, \(\sin \frac{3\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).

Step 2: Calculate \(\sin \frac{\pi}{6}\)

From the unit circle, \(\sin \frac{\pi}{6}\) equals \(\frac{1}{2}\).

Step 3: Solve \(\sin \frac{3\pi}{4} - \sin \frac{\pi}{6}\)

Now subtract the values found in Steps 1 and 2: \(\frac{\sqrt{2}}{2} - \frac{1}{2}\).

Step 4: Use Sum-to-Product Identity

For \(\sin A - \sin B\), we use the identity: \(\sin A - \sin B = 2\cos \frac{A + B}{2} \sin \frac{A - B}{2}\). With \(A = \frac{3\pi}{4}\) and \(B = \frac{\pi}{6}\), calculate: \(\frac{A+B}{2} = \frac{11\pi}{12}\) and \(\frac{A-B}{2} = \frac{\pi}{4}\).

Step 5: Calculate \(\sin \frac{7\pi}{12}\)

Use the angles \(A = \frac{3\pi}{4}\) and \(B = \frac{\pi}{6}\) to find \(\sin(A-B)\) using the result of Step 4. Plug \(\cos \frac{11\pi}{12}\) and \(\sin \frac{\pi}{4}\) into identity from Step 4:\(2 \cdot \left(-\frac{\sqrt{6}}{4}\right) \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{3} - 1}{2}\).

Step 6: Combine Results for Final Answer

For part (a), the calculated difference from Step 3 is \(\frac{\sqrt{2}}{2} - \frac{1}{2} = \frac{\sqrt{2} - 1}{2}\). For part (b), from Step 5 the exact value is \(\sin \frac{7\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}\).

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are mathematical formulas used to simplify the expressions involving the sum or difference of two trigonometric functions. Specifically, these identities can transform sums or differences into products, which are often easier to compute. They involve sine, cosine, and tangent functions. The main identity we're focusing on here is:

• For two angles, \(\sin A - \sin B = 2\cos \frac{A + B}{2} \sin \frac{A - B}{2}\)

To find values like \(\sin \frac{7\pi}{12}\), knowing these identities can be a powerful tool.
They help us convert complex trigonometric expressions into simpler forms, making them easier to evaluate without any approximation. Ultimately, mastering these identities gives a deeper understanding of trigonometry's structure.

Unit Circle

The unit circle is an essential concept in trigonometry. It is a circle with a radius of 1 centered at the origin of a Cartesian coordinate system. The unit circle's primary role is to define the trigonometric functions for all real numbers.
For instance, any point \((x, y)\) on this circle satisfies the equation \(x^2 + y^2 = 1\).
When we talk about angles in the unit circle, they are typically measured in radians.

• The sine of an angle is the y-coordinate of the corresponding point on the unit circle.
• Cosine is represented by the x-coordinate.

The simplicity of the unit circle allows us to easily visualize and compute the values of trigonometric functions at various angles. For example, we know that \(\sin \frac{\pi}{6} = \frac{1}{2}\) and \(\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}\), directly by examining the circle.

Trigonometric Functions

Trigonometric functions are fundamental in the study of triangles and oscillatory phenomena. These functions relate angles to the ratios of the sides of a right triangle. The primary functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).

• Sine function explains the relationship between an angle and the length of the opposite side in a right triangle relative to the hypotenuse.
• Cosine relates the angle to the length of the adjacent side over the hypotenuse.
• Tangent is the ratio of the sine and cosine of an angle.

These relationships aren't limited to triangle sides. They extend to any real number, often through movements on the unit circle. Understanding these functions involves grasping concepts like quadrants where each function can assume positive or negative values. This understanding is vital in solving problems involving angles like the ones in our exercise.

Angle Subtraction

Angle subtraction is an important concept in trigonometry, especially when dealing with trigonometric identities or transformations. It is about finding the sine, cosine, or tangent of an angle that is described by the difference of two other angles. For example, the formula for sine is:

• \(\sin (A - B) = \sin A \cos B - \cos A \sin B\)

When calculating angles like \(\sin \frac{7\pi}{12}\), the problem can be simplified using the subtraction of angles already known, such as \(\frac{3\pi}{4}\) and \(\frac{\pi}{6}\).
This approach utilizes known values and precise identities. Understanding this concept ensures proficiency in tackling more complex trigonometric problems involving angle transformations.