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Chapter 4: Problem 14

Find the four smallest positive numbers \(\theta\) such that \(\sin \theta=0\)

Short Answer

Expert verified
The four smallest positive values of \(\theta\) for which \(\sin \theta = 0\) are \(\theta=\pi\), \(2\pi\), \(3\pi\), and \(4\pi\).

Step by step solution

01

Identifying the sine function's zeros on the unit circle

Recall that the sine function measures the y-coordinate of a point in the unit circle. The sine function is equal to 0 when the point is on the x-axis, which corresponds to the angles 0, 蟺, 2蟺, 3蟺, ...
02

Find the first four positive values of \(\theta\) for which \(\sin \theta = 0\)

Since the sine function is zero for integral multiples of 蟺, we have the following cases: 1. For 1 multiple of 蟺, we have \(\theta = 1\cdot \pi\) 2. For 2 multiples of 蟺, we have \(\theta = 2\cdot \pi\) 3. For 3 multiples of 蟺, we have \(\theta = 3\cdot \pi\) 4. For 4 multiples of 蟺, we have \(\theta = 4\cdot \pi\) Thus, the four smallest positive values for the angle \(\theta\) for which \(\sin \theta = 0\) are: 1. \(\theta = \pi\) 2. \(\theta = 2\pi\) 3. \(\theta = 3\pi\) 4. \(\theta = 4\pi\)

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Key Concepts

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

Unit Circle
The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate plane. This circle is important because it allows us to define trigonometric functions for all angles, both positive and negative. Every point on the unit circle can be described using coordinates \(x, y\), where \(x = \cos{\theta}\) and \(y = \sin{\theta}\). This relationship tells us that the \(x\) coordinate corresponds to the cosine of the angle, and the \(y\) coordinate corresponds to the sine of the angle.
  • The angle \(\theta\) is measured starting from the positive x-axis.
  • Angles are measured in radians, where one full circle is \(2\pi\) radians.
  • The sine function is the projection of the radius onto the y-axis and ranges from -1 to 1.
A key feature of the unit circle is its symmetry, which helps in understanding properties such as the sine function's periodic nature.
Sine Function Zeros
The zeros of the sine function are the angles where the sine value equals zero. On the unit circle, this happens when the angle corresponds to points on the x-axis, because at these points, the y-coordinate is zero.
  • These angles are \(0, \pi, 2\pi, 3\pi, \, ...\)
  • The sine function is sinusoidal and periodic with period \(2\pi\).
  • This pattern repeats for every complete rotation around the circle.
This means that the sine of any integer multiple of \(\pi\) is zero. Understanding the pattern of zeros helps solve equations involving the sine function, such as determining angular positions where the value of sine equals zero.
Angular Multiples
An angular multiple simply refers to an angle that is a multiple of another, typically one of \(\pi\) in trigonometry. In the context of the sine function, if we look at multiples of \(\pi\), they tell us all the angles where \(\sin\theta = 0\).
  • If \((\theta = n \times \pi), \, \sin\theta = 0\) for any integer \(n\).
  • The smallest such angles are those where \(n\) is a small positive integer; \(n = 1, 2, 3, 4,\, ...\).
  • This provides a straightforward method to calculate angles resulting in specific sine values.
By recognizing these angular multiples, better understanding of trigonometric problems can be achieved, particularly in determining cycle completion points or identifying specific angle values where certain properties hold.

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Most popular questions from this chapter

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$ \tan u=-2 \text { and } \tan v=-3 $$ Find exact expressions for the indicated quantities. $$ \sin (-v) $$

Explain why \(|\cos (x+n \pi)|=|\cos x|\) for every number \(x\) and every integer \(n\).

(a) Show that $$ x^{3}+x^{2} y+x y^{2}+y^{3}=\left(x^{2}+y^{2}\right)(x+y) $$ for all numbers \(x\) and \(y\). (b) Show that $$ \begin{aligned} \cos ^{3} \theta+\cos ^{2} \theta \sin \theta &+\cos \theta \sin ^{2} \theta+\sin ^{3} \theta \\ &=\cos \theta+\sin \theta \end{aligned} $$ for every number \(\theta\).

Show that $$ \frac{\sin x}{1-\cos x}=\frac{1+\cos x}{\sin x} $$ for every number \(x\) that is not an integer multiple of \(\pi\).

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived.] $$ \sin \frac{13 \pi}{12} $$
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