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Chapter 5: Problem 22

If \(\sin \theta=0.3,\) determine the value of \(\sin \theta+\sin (\theta+2 \pi)+\sin (\theta+4 \pi)\)

Short Answer

Expert verified
0.9

Step by step solution

01

Understand the Sine Function Periodicity

The sine function, \(\text{sin} \theta\), is periodic with a period of \({2\text{蟺}}\). This means that \(\text{sin}(\theta + 2\text{蟺}) = \text{sin} \theta\) for any angle \(\theta\).
02

Apply Periodicity to Given Angles

Using the periodic property, we can write: \(\text{sin}(\theta + 2\text{蟺}) = \text{sin} \theta\) and \(\text{sin}(\theta + 4\text{蟺}) = \text{sin} \theta\).
03

Substitute and Simplify

Substitute \(\text{sin}(\theta + 2\text{蟺})\) and \(\text{sin}(\theta + 4\text{蟺})\) with \(\text{sin} \theta\): \(\text{sin} \theta + \text{sin}(\theta + 2\text{蟺}) + \text{sin}(\theta + 4\text{蟺}) = \text{sin} \theta + \text{sin} \theta + \text{sin} \theta = 3 \text{sin} \theta\).
04

Calculate the Final Value

Given that \(\text{sin} \theta = 0.3\), substitute this value into the equation: \(\text{sin} \theta + \text{sin}(\theta + 2\text{蟺}) + \text{sin}(\theta + 4\text{蟺}) = 3 \times 0.3 = 0.9\).

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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, often written as \(\text{sin} \theta\), is one of the basic trigonometric functions in mathematics. It describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. The sine function is essential in modeling periodic phenomena like sound waves and light waves.
Key properties:
  • The sine function ranges from -1 to 1, in its graph.
  • When \(\theta = 0\), \(\text{sin} \theta = 0\).
  • When \(\theta = \frac{\text{蟺}}{2}\), \(\text{sin} \theta = 1\).
  • Sine is an odd function: \(\text{sin}(-\theta) = -\text{sin}(\theta)\).

Understanding these properties helps in solving various trigonometric problems effectively.
Periodicity
Periodicity refers to the repeating nature of a function at regular intervals. For the sine function, the period is \({2\text{蟺}}\). This means that for any angle \(\theta\), \(\text{sin}(\theta + 2\text{蟺}) = \text{sin} \theta\).
In simpler terms:
  • If you add \({2\text{蟺}}\) to any angle, the sine value remains the same.
Let's apply periodicity to the given exercise:
Given that \(\text{sin} \theta = 0.3\), we have:
  • \(\text{sin}(\theta + 2\text{蟺}) = 0.3\)
  • \(\text{sin}(\theta + 4\text{蟺}) = 0.3\)
Combining these results:
\(\text{sin} \theta + \text{sin}(\theta + 2\text{蟺}) + \text{sin}(\theta + 4\text{蟺}) = 3 \text{sin} \theta = 3 \times 0.3 = 0.9\)
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. These identities are useful in simplifying expressions and solving equations.
Some key identities include:
  • Pythagorean identities: \(\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1\)
  • Angle sum and difference identities:
    \(\text{sin}(\theta \(\text{卤}\) \(\text{蠁}\)) = \text{sin} \theta \text{cos} \(\text{蠁}\) \(\text{卤}\) \text{cos} \theta \text{sin} \(\text{蠁}\)\)
  • Double-angle identities:
    \(\text{sin}(2\theta) = 2 \text{sin} \theta \text{cos} \theta\)
By using these identities, it becomes easier to deal with complex trigonometric problems.
In our exercise, we used the periodicity identity to solve for the value of \(\text{sin} \theta + \text{sin}(\theta + 2\text{蟺}) + \text{sin}(\theta + 4\text{蟺})\), demonstrating the importance of understanding these fundamental identities.

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Most popular questions from this chapter

a) State the five key points for \(y=\sin x\) that occur in one complete cycle from \(\mathbf{0}\) to \(2 \boldsymbol{\pi}\) b) Use the key points to sketch the graph of \(y=\sin x\) for \(-2 \pi \leq x \leq 2 \pi .\) Indicate the key points on your graph. c) What are the \(x\) -intercepts of the graph? d) What is the \(y\) -intercept of the graph? e) What is the maximum value of the graph? the minimum value?

A security camera scans a long straight fence that encloses a section of a military base. The camera is mounted on a post that is located \(5 \mathrm{m}\) from the midpoint of the fence. The camera makes one complete rotation in 60 s. a) Determine the tangent function that represents the distance, \(d\), in metres, along the fence from its midpoint as a function of time, \(t,\) in seconds, if the camera is aimed at the midpoint of the fence at \(t=0\) b) Graph the function in the interval \(-15 \leq t \leq 15\) c) What is the distance from the midpoint of the fence at \(t=10 \mathrm{s},\) to the nearest tenth of a metre? d) Describe what happens when \(t=15 \mathrm{s}\)

A mass attached to the end of a long spring is bouncing up and down. As it bounces, its distance from the floor varies sinusoidally with time. When the mass is released, it takes \(0.3 \mathrm{s}\) to reach a high point of 60 cm above the floor. It takes 1.8 s for the mass to reach the first low point of \(40 \mathrm{cm}\) above the floor. a) Sketch the graph of this sinusoidal function. b) Determine the equation for the distance from the floor as a function of time. c) What is the distance from the floor when the stopwatch reads \(17.2 \mathrm{s?}\) d) What is the first positive value of time when the mass is \(59 \mathrm{cm}\) above the floor?

Does \(y=\) tan \(x\) have an amplitude? Explain.

Sketch the graph of each function over the interval \(\left[-360^{\circ}, 360^{\circ}\right] .\) For each function, clearly label the maximum and minimum values, the \(x\) -intercepts, the \(y\) -intercept, the period, and the range. a) \(y=2 \cos x\) b) \(y=-3 \sin x\) c) \(y=\frac{1}{2} \sin x\) d) \(y=-\frac{3}{4} \cos x\)
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