Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \sin 4 \theta-\sin 8 \theta $$

Step-by-Step Solution

Answer

\(\sin 4\theta - \sin 8\theta = 2 \cos (6\theta) \sin (2\theta)\)

Step 1: Recall the Identity

First, we need to use the sine difference identity. The formula is:\[\ \sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)\]Apply this formula to \(\sin 4\theta - \sin 8\theta\).

Step 2: Identify A and B

For our problem, we can let \(A = 8\theta\) and \(B = 4\theta\).

Step 3: Calculate A+B and A-B

Calculate \(A+B\) and \(A-B\). \[A + B = 8\theta + 4\theta = 12\theta\]\[A - B = 8\theta - 4\theta = 4\theta\]

Step 4: Substitute and Simplify

Substitute these values into the sine difference identity:\[\sin 4\theta - \sin 8\theta = 2 \cos \left(\frac{12\theta}{2}\right) \sin \left(\frac{4\theta}{2}\right)\]Simplify the expression:\[= 2 \cos (6\theta) \sin (2\theta)\]This is the expression as a product.

Key Concepts

Understanding the Sine Difference Identity

The sine difference identity is a powerful tool used in trigonometry to simplify expressions involving the difference of two sine functions. It states that

• \( \sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \).

Although it might seem a bit complicated at first glance, its application is rather straightforward.

The purpose of this identity is to transform the difference of sines into a product of cosine and sine, simplifying calculations and revealing underlying relationships within an expression.
For example, when faced with \( \sin 4\theta - \sin 8\theta \), we can utilize this identity to express it in a different form that might be easier to work with in further calculations or proofs. Just identify \( A \) and \( B \), plug them into the formula, and simplify the expression.

Insights into the Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Cosine functions are periodic, with a period of \( 2\pi \) radians.

• Its range is between -1 and 1.
• It helps in determining the horizontal component of circular movement.

In the identity \( 2\cos \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right) \), the cosine component helps modulate or scale the resulting product, affecting its amplitude based on the inputs \( A \) and \( B \).

So, in our example, \( \cos(6\theta) \) is critical as it defines one aspect of the resulting product after applying the sine difference identity. Understanding this function is key for interpreting and solving various trigonometric problems.

Role of the Sine Function

The sine function is another fundamental trigonometric function commonly used in various mathematical and physical contexts. Like cosine, it is based on the right-angled triangle and can be also defined using the unit circle, representing the y-coordinate of the rotating point.

• Sine also has a range between -1 and 1.
• Its period is \( 2\pi \) radians.
• The sine function is crucial for determining the vertical component in circular motion.

In the expression \( 2\cos (6\theta) \sin (2\theta) \), the sine function \( \sin(2\theta) \) represents another component of the transformed expression.

By understanding the sine function, we can grasp how variations and transformations occur in trigonometric expressions. Knowing how sine interacts with other trigonometric functions, especially in identities like the sine difference, is vital for simplifying expressions and solving equations.