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Chapter 7: Problem 13

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

Short Answer

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

Step by step solution

01

Identify the Formula

To express the difference of two sines as a sum or difference, use the formula: \( \sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \).
02

Substitute Values into Formula

In the formula, let \( A = 7t \) and \( B = 3t \). Substituting these gives us: \( \sin 3t - \sin 7t = 2 \cos \left( \frac{7t + 3t}{2} \right) \sin \left( \frac{7t - 3t}{2} \right) \).
03

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left[2 \arccos \left(-\frac{3}{5}\right)\right]\) (b) \(\cos \left(2 \sin ^{-1} \frac{15}{17}\right)\) (c) \(\tan \left(2 \tan ^{-1} \frac{3}{4}\right)\)

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 1-\sin t=\sqrt{3} \cos t $$

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \cot \alpha+\tan \alpha=\csc \alpha \sec \alpha $$

Exer. 53-64: Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. \((\cos x)(15 \cos x+4)=3\) \([0,2 \pi)\)

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left(\sin ^{-1} \frac{2}{3}\right)\) (b) \(\cos \left[\cos ^{-1}\left(-\frac{1}{5}\right)\right]\) (c) \(\tan \left[\tan ^{-1}(-9)\right]\)
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