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Chapter 5: Problem 38

Reference Numbers Find the reference number for each value of \(t\) (a) \(t=9 \pi\) (b) \(t=-\frac{5 \pi}{4}\) (c) \(t=\frac{25 \pi}{6}\) (d) \(t=4\)

Short Answer

Expert verified
(a) 0; (b) \(\frac{\pi}{4}\); (c) \(\frac{\pi}{6}\); (d) \(0.858\).

Step by step solution

01

Understand Reference Numbers

Reference numbers are angles between 0 and \( \pi \) that help us understand the position of an angle in the unit circle. For any angle \( t \), the reference number \( t' \) is the shortest distance from \( t \) to the x-axis.
02

Find Reference Number for \( t=9\pi \)

Since \( 9\pi \) completes \( 4.5 \) full circles (i.e., \( 9\pi \div 2\pi = 4.5 \)), it lands on the starting axis, i.e., at \( \pi \) on the x-axis. Therefore, the reference number for \( t=9\pi \) is \( 0 \) since \( \pi \) is directly on the x-axis, and there is no vertical distance from the x-axis.
03

Find Reference Number for \( t=-\frac{5\pi}{4} \)

First, find the positive equivalent by adding \( 2\pi \): \( -\frac{5\pi}{4} + 2\pi = \frac{3\pi}{4} \). In the second quadrant, the reference number is calculated as \( \pi - \theta \), so \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).
04

Find Reference Number for \( t=\frac{25\pi}{6} \)

Divide \( 25\pi/6 \) by \( 2\pi \) to see how many full circles it makes: \( \frac{25\pi}{6} \div 2\pi = \frac{25}{12} \approx 2.083 \). This means 25\pi/6 makes 2 full cycles (2\times 2\pi) and \( 1\times \frac{\pi}{6} \) beyond. Thus, in the first quadrant, the reference number is directly \( \frac{\pi}{6} \).
05

Find Reference Number for \( t=4 \)

Convert 4 into radians: \( 4 \text{ rad} \approx 229.183 \text{ degrees} \). \( 4 \text{ rad} \) is in the third quadrant (since 3\pi/2 radians \( = 270 \text{ degrees} \) is close). The reference number is \( t - \pi \), so \( 4 - \pi \approx 0.858 \).

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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 that helps us understand the relationship between angles and the coordinates of points on a circle with a radius of one.
  • The unit circle is centered at the origin of a coordinate plane.
  • Any point on the unit circle can be described by the coordinates \( (\cos \theta, \sin \theta) \), where \( \theta \) is the angle formed with the positive x-axis.
  • This provides a visual way to connect angle measures with trigonometric values like sine and cosine.
Understanding the positions and quadrants on the unit circle assists with calculating reference numbers, as it allows us to determine where the angle lies and how it relates to the x-axis.
Radians
Radians are a unit of angular measure used in many areas of mathematics. Unlike degrees, radians express angles in terms of the arc length of a circle.
  • One complete circle in radians is \( 2\pi \) radians, equivalent to 360 degrees.
  • To convert degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
  • Radians provide a natural measure that simplifies many mathematical expressions and calculations.
Using radians allows us to easily integrate mathematical operations, especially when dealing with periodic functions like those found in trigonometry.
Angle Measurement
Trigonometry relies heavily on measuring angles, which can be expressed in degrees or radians, depending on the context.
  • Angles describe the rotation from the initial side to the terminal side in a plane.
  • Measuring in radians is often preferred in higher-level mathematics due to its simplicity and applicability. For example, the angle in radians shows a direct relation to the radius and arc length.
Understanding how to measure angles in both radians and degrees is crucial for converting between the two, a common task when working with reference numbers.
Trigonometry
Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles, as well as the properties of waves in the context of circles.
  • Key concepts in trigonometry include sine, cosine, and tangent functions, which are fundamentally tied to the unit circle.
  • Trigonometry helps solve problems involving right triangles and circular functions, making it vital in fields like physics and engineering.
  • Understanding reference numbers involves applying trigonometric principles to find the simplest equivalent angle within a circle.
Mastering trigonometry reinforces foundational skills that enable us to navigate various mathematical problems, including those involving the unit circle and reference numbers.

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Most popular questions from this chapter

The pendulum in a grandfather clock makes one complete swing every 2 s. The maximum angle that the pendulum makes with respect to its rest position is \(10^{\circ} .\) We know from physical principles that the angle \(\theta\) between the pendulum and its rest position changes in simple harmonic fashion. Find an equation that describes the size of the angle \(\theta\) as a function of time. (Take \(t=0\) to be a time when the pendulum is vertical.) (Figure cant copy)

When a car with its horn blowing drives by an observer, the pitch of the horn seems higher as it approaches and lower as it recedes (see the figure below). This phenomenon is called the Doppler effect. If the sound source is moving at speed \(v\) relative to the observer and if the speed of sound is \(v_{0}\), then the perceived frequency \(f\) is related to the actual frequency \(f_{0}\) as follows. $$f=f_{0}\left(\frac{v_{0}}{v_{0} \pm v}\right)$$ We choose the minus sign if the source is moving toward the observer and the plus sign if it is moving away. Suppose that a car drives at \(110 \mathrm{ft} / \mathrm{s}\) past a woman standing on the shoulder of a highway, blowing its horn, which has a frequency of \(500 \mathrm{Hz}\). Assume that the speed of sound is \(1130 \mathrm{ft} / \mathrm{s} .\) (This is the speed in dry air at \(70^{\circ} \mathrm{F}\).) (a) What are the frequencies of the sounds that the woman hears as the car approaches her and as it moves away from her? (b) Let \(A\) be the amplitude of the sound. Find functions of the form $$y=A \sin \omega t$$ that model the perceived sound as the car approaches the woman and as it recedes. (Figure cant copy)

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. $$\cos t, \sin t ; \quad \text { Quadrant IV }$$

Find the maximum and minimum values of the function. $$y=2 \sin x+\sin ^{2} x$$

Blood Pressure Each time your heart beats, your blood pressure first increases and then decreases as the heart rests between beats. The maximum and minimum blood pressures are called the systolic and diastolic pressures, respectively. Your blood pressure reading is written as systolic/diastolic. A reading of \(120 / 80\) is considered normal. A certain person's blood pressure is modeled by the function $$ p(t)=115+25 \sin (160 \pi t) $$ where \(p(t)\) is the pressure in \(\mathrm{mmHg}\) (millimeters of mercury), at time \(t\) measured in minutes. (a) Find the period of \(p\) (b) Find the number of heartbeats per minute. (c) Graph the function \(p\) (d) Find the blood pressure reading. How does this compare to normal blood pressure?
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