Problem 92 - Precalculus with Limits

Question

In Exercises 91-98, use the sum-to-product formulas to write the sum or difference as a product. \( \sin 5 \theta - \sin 3 \theta \)

Step-by-Step Solution

Answer

The expression \( \sin 5 \theta - \sin 3 \theta \) can be rewritten as \( 2 \cos 4 \theta \sin \theta \) using the sum-to-product formula.

Step 1: Identify the values of 'a' and 'b'

We have a trigonometric expression \( \sin 5 \theta - \sin 3 \theta \), in which two sine functions are subtracted. The coefficients of \( \theta \) in these sine functions are 'a' and 'b'. Therefore, in our problem, \( a = 5 \theta \) and \( b = 3 \theta \).

Step 2: Substitute 'a' and 'b' into the sum-to-product formula for sine functions

The sum-to-product formula for subtracted sine functions is \(\sin a - \sin b = 2 \cos \frac{a+b}{2} \sin \frac{a-b}{2}\). We can substitute \( a = 5 \theta \) and \( b = 3 \theta \) into this formula. This gives us: \(\sin 5 \theta - \sin 3 \theta = 2 \cos \frac{5 \theta+3 \theta}{2} \sin \frac{5 \theta-3 \theta}{2}\).

Step 3: Simplify the resulting expression

Now we can simplify this equation, which becomes: \(\sin 5 \theta - \sin 3 \theta = 2 \cos 4 \theta \sin \theta\). Thus the expression \( \sin 5 \theta - \sin 3 \theta \) has been rewritten as a product using the sum-to-product formula.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions, such as sine, cosine, and tangent, to each other. They are very useful tools in trigonometry, allowing us to simplify complex expressions, solve equations, and prove relationships among angles. Trigonometric identities can be used to convert sums or differences of angles into products and vice versa. This is particularly helpful because products can be easier to compute or manipulate in equations.

Here are some widely used trigonometric identities:

• Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle sum identity for sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• Angle sum identity for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• Sum-to-product identity for sine: \( \sin a \pm \sin b = 2\sin\left(\frac{a \pm b}{2}\right)\cos\left(\frac{a \mp b}{2}\right) \)

Using these identities, we can tackle the problem of converting the sine difference formula into a simpler product form. These transformations are foundational for solving trigonometric equations more elegantly.

Angle Subtraction

Angle subtraction is a method used in trigonometry to compute the sine or cosine of two angles when they are subtracted. This technique typically employs the angle subtraction identities, allowing for expressions involving sine or cosine of a difference of two angles to be broken down into more manageable operations.

For the sine function, we have:

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

The angle subtraction identity essentially breaks down the operation to use simpler trigonometric terms. This makes it easier to work with complex angle expressions, particularly when you need to collaborate multiple identities to simplify trigonometric equations. By reducing the complexity of the expression, angle subtraction aids in both computational efficiency and clarity in mathematical proofs.

Sine Function

The sine function is one of the most fundamental trigonometric functions. Defined for any angle \( \theta \), its value represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. The sine function, usually noted as \( \sin \theta \), is a periodic and continuous function.

• Period: \( 2\pi \)
• Range: [-1, 1]
• Key Identity: \( \sin(\pi/2 - \theta) = \cos \theta \)

The graph of \( \sin\) is a wave oscillating between -1 and 1. Moreover, it helps in modeling periodic phenomena like sound and light waves. In trigonometry, sine plays a key role in sum-to-product conversions, enabling the translation of sums or differences of angles into more soluble algebraic expressions. This function aids in solving equations involving angle measurements more seamlessly.

Cosine Function

The cosine function, another primary trigonometric function, relates the angle \( \theta \) of a right triangle to the lengths of the adjacent side and the hypotenuse. Defined as \( \cos \theta \), it, like sine, is periodic and continuous, affecting how angles and cyclical phenomena are understood mathematically.

• Period: \( 2\pi \)
• Range: [-1, 1]
• Key Identity: \( \cos(\pi/2 - \theta) = \sin \theta \)

The cosine function charts a similar wave to sine, creating a phase shift of \( \frac{\pi}{2} \) radians. In trigonometry, cosine assists in forming sum-to-product transformations, especially in transforming trigonometric sums or differences into products. These transformations often lead to solutions of trigonometric equations in more digestible forms. Cosine's role in the sum and difference identities further aids its function in understanding circuitry, vibrations, and waves.