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Chapter 7: Problem 81

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

Short Answer

Expert verified
0.9

Step by step solution

01

- Analyze the function's periodicity

The sine function \(\sin(\theta)\) is periodic with period \(2\pi\). This means that \(\sin(\theta + 2k\pi) = \sin(\theta)\) for any integer k. Thus, \(\sin(\theta + 2\pi) = \sin(\theta)\) and \(\sin(\theta + 4\pi) = \sin(\theta)\).
02

- Substitute known values

Given that \(\sin(\theta) = 0.3\), we can use the periodic property from Step 1: \(\sin(\theta + 2\pi) = 0.3\) and \(\sin(\theta + 4\pi) = 0.3\).
03

- Sum the terms

Now, sum the individual values: \(\sin(\theta) + \sin(\theta + 2\pi) + \sin(\theta + 4\pi) = 0.3 + 0.3 + 0.3 = 0.9\).

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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 is one of the fundamental trigonometric functions. It is defined for all real numbers and assigns an output value that cycles between -1 and 1.

The function is denoted as \(\sin(\theta)\), where \(\theta\) is the angle in radians.

Key properties of the sine function include:
  • It is periodic, which means it repeats its values at regular intervals.
  • The highest value of \(\sin(\theta)\) is 1, and the lowest value is -1.
  • Its graph is a smooth, wave-like curve called a sine wave.
Understanding the sine function is crucial as it forms the basis for analyzing more complex trigonometric functions and relationships.
Periodicity
Periodicity refers to the repeating nature of certain functions. For the sine function, this concept is especially important.

The sine function repeats its values every \[2\pi\]. This means:
  • The graph of \(\sin(\theta)\) looks the same after every interval of \[2\pi\] radians.
  • We say that the sine function has a period of \[2\pi\].
This periodic nature helps solve problems where angles are large or exceed \[2\pi\]. Instead of working with large angles, one can use equivalent angles within the \[0, 2\pi\] range. As seen in the exercise, \(\sin(\theta + 2\pi)\) and \(\sin(\theta + 4\pi)\) are both equal to \(\sin(\theta)\), simplifying calculations significantly.
Trigonometric Functions
Trigonometric functions, including sine, cosine, and tangent, describe relationships between angles and sides of triangles.

They are essential in various fields such as physics, engineering, and computer graphics.

Key trigonometric functions include:
  • \(\sin(\theta)\): Determines the ratio of the opposite side to the hypotenuse in a right triangle.
  • \(\cos(\theta)\): The ratio of the adjacent side to the hypotenuse.
  • \(\tan(\theta)\): The ratio of the opposite side to the adjacent side.
Additionally, these functions are periodic and have specific ranges of values they cycle through. This exercise, focusing on the sine function and its periodicity, is a great example of applying these fundamental principles efficiently.

Being familiar with trigonometric functions and their properties aids in solving various mathematical problems.

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Most popular questions from this chapter

Use a calculator to find the approximate value of each expression. Round the answer to two decimal places. $$ \tan 36^{\circ} $$

\(f(x)=\sin x, g(x)=\cos x, h(x)=2 x,\) and \(p(x)=\frac{x}{2} .\) Find the value of each of the following: $$ (f \cdot g)\left(\frac{\pi}{4}\right) $$

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\sec 35^{\circ} \csc 55^{\circ}-\tan 35^{\circ} \cot 55^{\circ}$$

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\tan 10^{\circ} \cot 10^{\circ}$$

Problems \(63-66\) require the following discussion. Projectile Motion The path of a projectile fired at an inclination \(\theta\) to the horizontal with initial speed \(v_{0}\) is a parabola. See the figure. The range \(R\) of the projectile-that is, the horizontal distance that the projectile travels-is found by using the function $$ R(\theta)=\frac{2 v_{0}^{2} \sin \theta \cos \theta}{g} $$ where \(g \approx 32.2\) feet per second per second \(\approx 9.8\) meters per second per second is the acceleration due to gravity. The maximum height \(H\) of the projectile is given by the function $$ H(\theta)=\frac{v_{0}^{2} \sin ^{2} \theta}{2 g} $$ Find the range \(R\) and maximum height \(H\) of the projectile. Round answers to two decimal places. The projectile is fired at an angle of \(45^{\circ}\) to the horizontal with an initial speed of 100 feet per second.
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