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Magazine Chapter 8: Problem 95
Find the real zeros of each trigonometric function on the interval \(0 \leq \theta<2 \pi\) \(f(x)=\sin (2 x)-\sin x\)
Short Answer
Expert verified
The real zeros are \(0, \pi/3, \pi, 5\pi/3\).
Step by step solution
01
Set the function equal to zero
To find the real zeros, we set the function equal to zero: \(\text{f(x) = \sin (2x) - \sin x = 0}\)
02
Use trigonometric identities
Recall the double angle identity: \(\text{sin (2x) = 2 sin(x) cos(x)}\). Substitute this into the equation: \(\text{2 sin(x) cos(x) - sin(x) = 0}\)
03
Factor the equation
Factor out \(\text{sin(x)}\): \(\text{sin(x)(2 cos(x) - 1) = 0}\)
04
Solve each factor equal to zero
Set each factor equal to zero and solve: \(\text{sin(x) = 0}\) \(\text{2 cos(x) - 1 = 0}\).
05
Solve \(sin(x) = 0\)
The solutions on the interval \(0 \leq \theta < 2 \pi\) are \(x = 0, \pi\).
06
Solve \(2 cos(x) - 1 = 0\)
Rewrite the equation: \(cos(x) = 1/2\). The solutions on the interval \(0 \leq \theta < 2 \pi\) are \(x = \pi/3, 5\pi/3\).
07
Combine all solutions
Combine all the solutions found in Steps 5 and 6: \(x = 0, \pi/3, \pi, 5\pi/3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Identities
Trigonometric identities play a crucial role in simplifying and solving trigonometric equations. These identities are mathematical statements that hold true for all values of the variables involved. Here are some common trigonometric identities:
- Pythagorean identities: \text{ (e.g., \(\text{sin}^2(x) + \text{cos}^2(x) = 1\)}
- Reciprocal identities: \text{ (e.g., \(\text{sec}(x) = 1/\text{cos}(x)\)}
Solving Trigonometric Equations
Solving a trigonometric equation involves finding all the angles that satisfy the equation within a given interval. Here are the general steps:
- Set the equation to zero: This helps isolate the trigonometric functions.
- Apply relevant trigonometric identities: Simplify the equation using identities.
- Factor the equation: Break it down into simpler parts.
- Solve each factor: Find the angles that satisfy each part individually.
Double Angle Identity
The double angle identity is a specific type of trigonometric identity used to express functions involving double angles. The double angle identities include:
- \(\text{sin}(2x) = 2\text{sin}(x)\text{cos}(x)\)
- \(\text{cos}(2x) = \text{cos}^2(x) - \text{sin}^2(x)\)
- \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\)
Factoring Trigonometric Expressions
Factoring is a fundamental algebraic technique used to solve trigonometric equations. It involves expressing a complex trigonometric expression as a product of simpler factors. Here are the steps involved:
- Identify common factors: Look for common trigonometric terms in the equation.
- Factor out the common term: Simplify the expression by extracting the common term.
- Solve each factor: Set each factor equal to zero and solve.
