Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \sin 3 t-\sin 7 t $$

Step-by-Step Solution

Answer

\(2 \cos 5t \sin 2t\)

Step 1: Identify the Formula

To express the difference of two sines as a sum or difference, use the formula: \( \sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \).

Step 2: Substitute Values into Formula

In the formula, let \( A = 7t \) and \( B = 3t \). Substituting these gives us: \( \sin 3t - \sin 7t = 2 \cos \left( \frac{7t + 3t}{2} \right) \sin \left( \frac{7t - 3t}{2} \right) \).

Step 3: Simplify the Formula

Calculate the expressions inside the cosine and sine functions: \( \frac{7t + 3t}{2} = 5t \) and \( \frac{7t - 3t}{2} = 2t \). Thus, the expression becomes: \( 2 \cos 5t \sin 2t \).

Key Concepts

Sum and difference formulas

When dealing with trigonometric identities, a particularly useful set of formulas are the sum and difference formulas. These formulas enable us to transform expressions involving the sums or differences of angles into different forms that can be easier to work with.

• The difference formula for sine is: \( \sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \).
• This formula can help simplify expressions or solve equations by expressing the sine difference in terms of cosine and sine.
• These identities are useful in trigonometric integrals and simplifying expressions in physics and engineering.

By using these formulas, complex expressions can be rendered into simpler parts, aiding in the computation of angles that are otherwise not straightforward.

Sine function

The sine function, denoted as \( \sin \), is one of the fundamental trigonometric functions related to right-angled triangles. In a unit circle, which has a radius of one unit, sine represents the y-coordinate of a point that forms an angle \( \theta \) with the positive x-axis.

• The function is periodic with a period of \( 2\pi \), meaning its values repeat every \( 2\pi \) intervals.
• The sine function varies between -1 and 1, inclusive, which defines its range.
• It is an odd function, which translates mathematically to \( \sin(-x) = -\sin(x) \), showing its symmetric property about the origin.

Understanding these properties is crucial when applying the sine function in real-world scenarios like sound waves, alternating currents, and cyclical patterns.

Cosine function

The cosine function, written as \( \cos \), is another principal trigonometric function closely associated with the unit circle. It represents the x-coordinate of a point on the circle corresponding to a given angle \( \theta \).

• Like sine, cosine is also periodic with the period being \( 2\pi \).
• The function ranges between -1 and 1, making it consistent in measurement.
• Cosine is an even function, represented mathematically as \( \cos(-x) = \cos(x) \), demonstrating its symmetry about the y-axis.

These properties allow the cosine function to be applied in various mathematical computations and practical applications, including in phasor studies in electrical engineering, and analyzing harmonic motion.