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Chapter 6: Problem 28

\(\sin \theta=0 \quad\) when \(\theta=n \pi\)

Short Answer

Expert verified
\( \theta = n \pi \).

Step by step solution

01

- Understand the Equation

We are given the trigonometric equation \(\text{sin} \theta = 0\). The solutions to this equation occur at specific values of \(\theta\).
02

- Recall the Sine Function

The sine function, \( \text{sin} \theta \), equals zero at integer multiples of \( \pi \). Therefore, \( \text{sin} \theta = 0 \) when \(\theta = n \pi \), where \(n\) is an integer.
03

- Conclusion

Combining the observations, we conclude that the general solution to \( \text{sin} \theta = 0 \) is \( \theta = n \pi \), where \(n\) is any integer.

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Key Concepts

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

sine function
The sine function, written as \(\text{sin}(\theta)\), is one of the fundamental trigonometric functions. It describes the relationship between the angles and sides of a right-angled triangle. The sine of an angle \(\theta\) is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. The sine function has some crucial properties: it oscillates between -1 and 1, and it is periodic with a period of \(2\text{蟺}\). This means that \(\text{sin}(\theta + 2\text{蟺}) = \text{sin}(\theta)\) for any angle \(\theta\).
trigonometric equations
Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. Solving these equations means finding all values of the variable that make the equation true. For instance, \(\text{sin}(\theta) = 0\) is a trigonometric equation. To solve it, we need to find all angles \(\theta\) that satisfy this condition. Trigonometric equations can have infinitely many solutions because the trigonometric functions are periodic.
general solutions of trigonometric equations
When solving trigonometric equations, we often seek a 'general solution' that captures all possible solutions. For the equation \(\text{sin}(\theta) = 0\), the general solution takes the form \(\theta = n \text{蟺}\), where \(n\) is any integer. This is because the sine function equals zero at every integer multiple of \(\text{蟺}\). Thus, by specifying \(n\) as any integer, we account for all angles \(\theta\) where \(\text{sin}(\theta) = 0\). This method of representing solutions ensures we include all periodic repetitions of the angle.
integer multiples of pi
Integer multiples of \(\text{蟺}\) play a significant role in trigonometry, especially when it comes to the sine and cosine functions. For \(\text{sin}(\theta) = 0\), the solutions are \(\theta = n \text{蟺}\), where \(n\) is an integer. This is because the sine function hits zero at angles such as \(0, \text{蟺}, 2\text{蟺}, -\text{蟺}\), and so on. By expressing these solutions as \(n \text{蟺}\), we generalize the result and include both positive and negative multiples, covering all possible cases. This understanding is key to solving many trigonometric problems.

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Most popular questions from this chapter

The graph of \(f(\theta) \equiv \cos \theta\) compared with the graph of \(f(\theta) \equiv \sin \theta\) is: (a) inverted (c) \(90^{\circ}\) to the right (b) \(90^{\circ}\) to the left (d) of equal period.

Explain briefly what is meant by: (a) a periodic function, (b) the amplitude of a sine wave, (c) the frequency of a circular function, (d) the general solution of a trigonometric equation, (e) a radian.

(a) The graph of a trigonometric function \(f(\theta)\) is continuous. (b) \(f(\theta) \equiv \cot \theta\).

The graph of the function \(f(\theta) \equiv \cos \left(2 \theta-\frac{\pi}{2}\right)\) has a period (a) \(2 \pi\) (b) \(\pi\) (c) \(\frac{\pi}{2}\) (d) \(-\frac{\pi}{2}\) (e) none of these.

Find the general solutions of the equations:\(\tan 5 \theta=\cot 2 \theta\)
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