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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

Copy and complete the table of properties for \(y=\sin x\) and \(y=\cos x\) for all real numbers. $$\begin{array}{l|l|l|} \text { Property } & y=\sin x & y=\cos x \\ \hline \text { maximum } & & \\ \hline \text { minimum } & & \\ \hline \text { amplitude } & & \\ \hline \text { period } & & \\ \hline \text { domain } & & \\ \hline \text { range } & & \\ \hline y \text { -intercept } \\ \hline x \text { -intercepts } \\ \hline \end{array}$$

The Canadian National Historic Windpower Centre, at Etzikom, Alberta, has various styles of windmills on display. The tip of the blade of one windmill reaches its minimum height of \(8 \mathrm{m}\) above the ground at a time of 2 s. Its maximum height is \(22 \mathrm{m}\) above the ground. The tip of the blade rotates 12 times per minute. a) Write a sine or a cosine function to model the rotation of the tip of the blade. b) What is the height of the tip of the blade after \(4 \mathrm{s} ?\) c) For how long is the tip of the blade above a height of \(17 \mathrm{m}\) in the first \(10 \mathrm{s} ?\)

a) Copy each equation. Fill in the missing values to make the equation true. i) \(4 \sin \left(x-30^{\circ}\right)=4 \cos (x-\square)\) ii) \(2 \sin \left(x-\frac{\pi}{4}\right)=2 \cos (x-\square)\) iii) \(-3 \cos \left(x-\frac{\pi}{2}\right)=3 \sin (x+\square)\) iv) \(\cos (-2 x+6 \pi)=\sin 2(x+\square)\) b) Choose one of the equations in part a and explain how you got your answer.

State the period for each periodic function, in degrees and in radians. Sketch the graph of each function. a) \(y=\sin 4 \theta\) b) \(y=\cos \frac{1}{3} \theta\) c) \(y=\sin \frac{2}{3} x\) d) \(y=\cos 6 x\)

Is the function \(f(x)=5 \cos x+3 \sin x\) sinusoidal? If it is sinusoidal, state the period of the function.
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