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Chapter 10: Problem 10

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\sin (-2.5)$$

Short Answer

Expert verified
\( \sin(-2.5) = -\sin(2.5) \)

Step by step solution

01

Identify the Even-Odd Identity

The sine function is an odd function. The identity for sine is \( \sin(-x) = -\sin(x) \). This means that the sine of a negative angle is the negative of the sine of the positive angle.
02

Apply the Identity

We use the odd identity to rewrite \( \sin(-2.5) \) as \( -\sin(2.5) \). This transformation allows you to express the sine of a negative angle as a simple transformation of the sine of a positive angle.

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Key Concepts

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

Even-Odd Identities
Even-Odd Identities are fundamental concepts in trigonometry that help us understand how trigonometric functions behave with negative angles. These identities can make calculations easier by converting negative angle expressions into their positive counterparts.
For trigonometric functions:
  • Even functions mean that the function's graph is symmetrical about the y-axis.
  • Odd functions mean that the function's graph is symmetrical about the origin.
The sine function is a classic example of an odd function, which we will dive into next. Recognizing these identities can simplify problems, especially when dealing with negative inputs.
Sine Function
The Sine Function is one of the primary functions in trigonometry and is essential for describing oscillatory phenomena. It's defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.
The sine function has specific characteristics:
  • It is periodic with a period of \(2\pi\).
  • It ranges between -1 and 1.
  • It is an odd function, which means \(\sin(-x) = -\sin(x)\).
Understanding these properties helps manipulate and transform expressions more easily, especially when negative angles are involved.
Negative Angles
Negative angles in trigonometry refer to the rotation direction of an angle, measured clockwise from the positive x-axis, which contrasts with positive angles measured counterclockwise.
Working with negative angles is crucial as they frequently appear in various mathematical and physical contexts. By using even-odd identities:
  • Negative angles can often be expressed as positive angles.
  • This expression simplifies the solution while retaining the original value and direction of the angle's measure.
This understanding is critical for tasks such as graphing or solving trigonometric equations.
Trigonometric Functions
Trigonometric Functions are the core elements of trigonometry and include sine, cosine, tangent, among others. They define relationships between the angles and sides of triangles and extend to any real numbers.These functions are critical for studying cycles and oscillations, such as sound waves, and circles. Each trigonometric function has unique properties:
  • Sine and cosine are periodic and range from -1 to 1.
  • Tangent has a periodicity of \(\pi\), differing from sine and cosine.
  • Each function has its specific even or odd nature that helps transform negative angles.
Mastering these functions enables solving complex problems involving periodicity, waves, and oscillations.

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Most popular questions from this chapter

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2 \sqrt{3} \cos \frac{x}{2}=-3$$

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{1}{\sec t-1}+\frac{1}{\sec t+1}=2 \cot t \csc t$$

Suppose you are solving a trigonometric equation to find solutions in \(\left[0^{\circ}, 360^{\circ}\right)\) and your work leads to $$ \frac{1}{3} \theta=45^{\circ}, 60^{\circ}, 75^{\circ}, 90^{\circ} $$ What are the corresponding values of \(\theta ?\)

Sound Waves Sound is a result of waves applying pressure to a person's eardrum. For a pure sound wave radiating outward in a spherical shape, the trigonometric function $$P=\frac{a}{r} \cos \left(\frac{2 \pi r}{\lambda}-c t\right)$$ can be used to model the sound pressure \(P\) at a radius of \(r\) feet from the source, where \(t\) is time in seconds, \(\lambda\) is length of the sound wave in feet, \(c\) is speed of sound in feet per second, and \(a\) is maximum sound pressure at the source measured in pounds per square foot. (Source: Beranek, L.., Noise and Vibration Control, Institute of Noise Control Engineering. Washington, DC.) Let \(\lambda=4.9\) feet and \(c=1026\) feet per second. (IMAGE CANNOT COPY) (a) Let \(a=0.4\) pound per square foot. Graph the sound pressure at a distance \(r=10\) feet from its source over the interval \(0 \leq t \leq 0.05 .\) Describe \(P\) at this distance. (b) Now let \(a=3\) and \(t=10 .\) Graph the sound pressure for \(0 \leq r \leq 20 .\) What happens to the pressure \(P\) as the radius \(r\) increases? (c) Suppose a person stands at a radius \(r\) so that $$r=n \lambda$$ where \(n\) is a positive integer. Use the difference identity for cosine to simplify \(P\) in this situation.

Solve each problem. Back Stress If a person bends at the waist with a straight back, making an angle of \(\theta\) degrees with the horizontal, then the force \(F\) exerted on the back muscles can be modeled by the equation $$F=\frac{0.6 W \sin \left(\theta+90^{\circ}\right)}{\sin 12^{\circ}}$$ where \(W\) is the weight of the person. (Source: Metcalf, H., Topics in Classical Biophysics, Prentice- Hall.) (a) Calculate \(F\) when \(W=170\) pounds and \(\theta=30^{\circ}\) (b) Use an identity to show that \(F\) is approximately equal to \(2.9 \mathrm{W} \cos \theta\) (c) For what value of \(\theta\) is \(F\) maximum?
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