/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Solve the given equation. $$\c... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 7

Solve the given equation. $$\cos \theta=-1$$

Short Answer

Expert verified
\( \theta = \pi \).

Step by step solution

01

Identify the Range of 胃

The cosine function, \( \cos \theta \), is periodic and defined for all real numbers. However, solutions are often sought in the interval \([0, 2\pi]\) when dealing with a single cycle. So, we will find \( \theta \) such that \( \theta \in [0, 2\pi] \).
02

Analyze the Cosine Function

The function \( \cos \theta \) is known for its properties. A key property is that \( \cos \theta = -1 \) at the point where \( \theta \) aligns directly opposite to the value where \( \cos \theta = 1 \) (which is at 0 for one cycle).
03

Determine the Specific 胃 Value

Knowing that \( \cos \theta = -1 \) occurs when \( \theta = \pi \), within the interval \([0, 2\pi]\). This is because the cosine function has a value of -1 at an angle of \( 180^\circ \), which is \( \pi \) radians.
04

Verify the Solution

To confirm the solution, substitute \( \theta = \pi \) back into the equation and check: \( \cos \pi = -1 \). This satisfies the original equation \( \cos \theta = -1 \). Therefore, \( \theta = \pi \) is the correct solution.

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Key Concepts

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

Cosine Function
The cosine function, denoted as \( \cos(\theta) \), is a fundamental part of trigonometry. It is one of the primary six trigonometric functions, which relate angles to ratios of sides in right-angled triangles. Here are some key points about the cosine function:

  • The cosine function takes an angle as input and outputs a value between -1 and 1. This output corresponds to the x-coordinate of a point on the unit circle that is \( \theta \) radians away from the positive x-axis.
  • Cosine is periodic with a period of \( 2\pi \). This means the function repeats its values every \( 2\pi \) radians.
  • At \( \theta = 0 \), \( \cos(\theta) = 1 \), while at \( \theta = \pi \), \( \cos(\theta) = -1 \). This is crucial when understanding symmetrical properties of the function.
The symmetry and periodicity of the cosine function play crucial roles in solving trigonometric equations, such as \( \cos(\theta) = -1 \). Understanding these properties allows us to find precise angles that satisfy the equation.
Radian Measure
Radian measure is another way to quantify angles, different from the more commonly known degree measure. This system is widely used in mathematics for several reasons:

  • One radian is defined as the angle created when the radius of a circle is wrapped around its circumference. Therefore, the circumference of a circle in radians is \( 2 \pi \) since the full circle is \( 2\pi \) times the radius.
  • Radians allow for simpler calculations in calculus and other fields of math. Trigonometric functions like sine and cosine work with radians in a more natural and concise way than degrees.
  • Key angles in radians are often used in trigonometry, like \( \pi/2 \), \( \pi \), and \( 3\pi/2 \), corresponding to 90掳, 180掳, and 270掳 respectively.
By working in radians, we can easily connect angles, arc lengths, and areas of sectors. In equations like \( \cos(\theta) = -1 \), \( \theta = \pi \) is the angle in radians where this equation holds true.
Unit Circle
The unit circle is a powerful visualization tool in trigonometry. It is a circle with a radius of 1 centered at the origin of the coordinate system.

  • Each point on the unit circle represents a possible angle \( \theta \). The coordinates of the point give us \( (\cos(\theta), \sin(\theta)) \).
  • The unit circle helps visualize the cosine and sine functions since angles rotate counterclockwise from the positive x-axis. For example, a 180掳 angle or \( \pi \) radians is at the point \((-1, 0)\).
  • As the angle changes around the circle, both cosine and sine values trace out wave-like paths. This illustrates their periodic and oscillatory nature.
Using the unit circle simplifies the identification of angles corresponding to given sine or cosine values. Specifically, \( \cos(\theta) = -1 \) is reached precisely at \( \theta = \pi \), as this is where the point on the unit circle is directly left along the x-axis.
Angle Measurement
Measuring angles can be approached in different ways, most notably degrees and radians. Understanding these measurements is crucial for solving equations involving angles.

  • Degrees divide a circle into 360 parts. This was historically used and is still prevalent in navigation and various branches of science.
  • Radians, however, relate angle measures directly to the arc lengths on a unit circle, offering benefits in mathematical computations and theory.
  • Conversion between degrees and radians is straightforward but essential: \( 180掳 \) equals \( \pi \) radians, and generally, degrees can be converted to radians by multiplying by \( \pi/180 \).
In problems like finding \( \theta \) for \( \cos(\theta) = -1 \), understanding angle measurement in radians helps identify accurate results efficiently. Knowing \( \theta = \pi \) results in \( \cos(\theta) = -1 \) allows us to examine the properties of these numbers in various trigonometric contexts.

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Most popular questions from this chapter

Sound Beats When two pure notes that are close in frequency are played together, their sounds interfere to produce beats; that is, the loudness (or amplitude) of the sound alternately increases and decreases. If the two notes are given by $$f_{1}(t)=\cos 11 t \quad \text { and } \quad f_{2}(t)=\cos 13 t$$ the resulting sound is \(f(t)=f_{1}(t)+f_{2}(t)\) (a) Graph the function \(y=f(t)\) (b) Verify that \(f(t)=2 \cos t \cos 12 t\) (c) Graph \(y=2 \cos t\) and \(y=-2 \cos t,\) together with the graph in part (a), in the same viewing rectangle. How do these graphs describe the variation in the loudness of the sound?

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$f(x)=\cos ^{2} x-\sin ^{2} x, \quad g(x)=1-2 \sin ^{2} x$$

Prove the identity. $$\frac{\sin 3 x+\sin 7 x}{\cos 3 x-\cos 7 x}=\cot 2 x$$

(a) Graph \(f(x)=\cos 2 x+2 \sin ^{2} x\) and make a conjecture. (b) Prove the conjecture you made in part (a).

Total Internal Reflection When light passes from a more-dense to a less-dense medium- from glass to air, for example- the angle of refraction predicted by Snell's Law (sce Exercise 57 ) can be \(90^{\circ}\) or larger. In this case the light beam is actually reflected back into the denser medium. This phenomenon, called total internal reflection, is the principle behind fiber optics. Set \(\theta_{2}=90^{\circ}\) in Snell's Law, and solve for \(\theta_{1}\) to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. (Note that the index of refraction from glass to air is the reciprocal of the index from air to glass.)
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