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Chapter 20: Problem 12

Simplify the given expressions. $$\sin 3 x \cos x-\sin x \cos 3 x$$

Short Answer

Expert verified
The simplified expression is \( \sin 2x \).

Step by step solution

01

Apply Trigonometric Identity

The problem can be solved using the identity for the difference of angles in sine: \[ \sin A \cos B - \sin B \cos A = \sin(A - B) \]Here, let \( A = 3x \) and \( B = x \). Thus, the expression \( \sin(3x)\cos(x) - \sin(x)\cos(3x) \) can be rewritten as: \[ \sin((3x) - x) = \sin(2x) \]
02

Conclusion

Therefore, the simplified form of the expression \( \sin 3x \cos x - \sin x \cos 3x \) is \( \sin 2x \).

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Key Concepts

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

Simplification of expressions
Simplification is a process to make mathematical expressions more manageable. We reduce complexity while retaining the original value.
  • In simplification, we often use known formulas and identities to reduce expressions.
  • This can make it easier to work with, especially in further calculations or analyses.

In our problem, we use a specific trigonometric identity to rewrite the initial expression. The identity \[ \sin A \cos B - \sin B \cos A = \sin(A - B) \] is employed. This identity helps transform the expression into a simpler form, showing how known identities streamline computations.
Angle subtraction formulas
The angle subtraction formulas in trigonometry allow us to express the sine, cosine, and tangent of the difference between two angles in terms of the trigonometric functions of the individual angles. This is especially useful when dealing with complex trigonometric expressions.
  • These formulas are part of a broader set known as the angle addition and subtraction identities.
  • They are crucial in simplifying expressions and solving equations involving trigonometric functions.

In the given exercise, the angle subtraction formula for sine: \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \] is applied. By setting \( A = 3x \) and \( B = x \), we effectively consolidated the expression into \( \sin(2x) \). Understanding this concept can significantly help in solving similar problems.
Sine function properties
The sine function is one of the primary trigonometric functions alongside cosine and tangent. Recognizing its properties is vital in working with trigonometric equations and simplifying expressions.
  • The sine function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) units.
  • Sine is an odd function, which means that \(\sin(-x) = -\sin(x)\).
  • It varies between -1 and 1, and is symmetric about the origin.

In trigonometric identities, the sine function often features due to its properties. While solving the exercise, it is crucial to leverage these inherent characteristics, ensuring an understanding of the function's behavior and values. This awareness allows for more intuitive problem-solving when using trigonometric identities for simplification.

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Most popular questions from this chapter

Solve the given problems. The equation for the trajectory of a missile fired into the air at an angle \(\alpha\) with velocity \(v_{0}\) is \(y=x \tan \alpha-\frac{g}{2 v_{0}^{2} \cos ^{2} \alpha} x^{2} .\) Here, \(g\) is the acceleration due to gravity. On the right of the equal sign, combine terms and simplify.

Solve the given problems. Simplify: \(\sin \left(\frac{\pi}{6}-\theta\right)+\cos \left(\frac{\pi}{3}-\theta\right)\)

Use a calculator to verify the given identities by comparing the graphs of each side. $$\cos y(\sec y-\cos y)=\sin ^{2} y$$

Solve the given problems. The current (in \(\Lambda\) ) in a certain \(A C\) circuit is given by \(i=4 \sin \left(120 \pi t+\frac{\pi}{2}\right) .\) Use the sum formula for sine to write this in a different form and then simplify.

Solve the given problems. In the study of the stress at a point in a bar, the equation \(s=a \cos ^{2} \theta+b \sin ^{2} \theta-2 t \sin \theta \cos \theta\) arises. Show that this equation can be written as \(s=\frac{1}{2}(a+b)+\frac{1}{2}(a-b) \cos 2 \theta-t \sin 2 \theta\).
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