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Chapter 7: Problem 2

Use identities to fill in the blanks. $$\text { If } \cos \theta=-0.65, \text { then } \cos (-\theta)=$$ ________

Short Answer

Expert verified
\( \cos(-\theta) = -0.65 \)

Step by step solution

01

Understanding the Properties of Cosine

Recall that cosine is an even function. This means that \( \cos(-\theta) = \cos(\theta) \). This property will help fill in the blank.
02

Substitute the Given Value

Substitute the given value of \( \cos(\theta) \) into the expression. Since \( \cos(\theta) = -0.65 \), it follows that \( \cos(-\theta) = -0.65 \).

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Key Concepts

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

even functions
In mathematics, an even function is one that satisfies the condition \(f(-x) = f(x)\) for every \(x\) in its domain. This symmetry means that the graph of the function is unchanged when reflected over the y-axis. Even functions are extremely useful in various branches of mathematics and physics.
Examples of even functions include:
  • The cosine function, \(\text{cos}(x)\). This is because \(\text{cos}(-x) = \text{cos}(x)\).
  • Quadratic functions like \(f(x) = x^2\).
  • Polynomials of the form \(a_0 + a_2x^2 + a_4x^4 + \text{...}\), where all exponents are even numbers.

Understanding even functions can simplify many problems, as it allows us to determine the function's behavior for negative inputs based on positive inputs alone.
cosine properties
The cosine function has several unique properties that make it a fundamental trigonometric function. One of the key properties of \(\text{cos}(x)\) is that it is an even function, which means \(\text{cos}(-x) = \text{cos}(x)\). This characteristic plays an essential role in solving trigonometric problems by providing symmetry.
Other important properties of the cosine function include:
  • The range of the cosine function is \([-1, 1]\).
  • The cosine function has a period of \(2\text{蟺}\), meaning \( \text{cos}(x + 2\text{蟺}) = \text{cos}(x)\).
  • It reaches its maximum value of 1 at \(x = 2k\text{蟺}\) and its minimum value of -1 at \(x = (2k+1)\text{蟺}\), where \(k\) is an integer.
  • Cosine is related to the unit circle, where \(\text{cos}(胃)\) represents the x-coordinate of the point on the unit circle corresponding to an angle 胃.

These properties make the cosine function extremely useful for describing oscillatory phenomena, waves, and alternating currents.
trigonometric functions
Trigonometric functions are functions that relate angles in right-angled triangles to the ratios of two sides of the triangle. They are essential in fields like geometry, physics, engineering, and even in finance.
Key trigonometric functions include:
  • Sin (Sine): \(\text{sin}(胃) = \frac{\text{opposite}}{\text{hypotenuse}}\)
  • Cos (Cosine): \(\text{cos}(胃) = \frac{\text{adjacent}}{\text{hypotenuse}}\). Cosine is fundamental for calculating horizontal components in physics.
  • Tan (Tangent): \(\text{tan}(胃) = \frac{\text{opposite}}{\text{adjacent}}\). Tangent is highly useful in various calculations, especially slope.

Besides these, there are three other trigonometric functions which are co-functions of the primary functions:
  • Cotangent (Cot): \(\text{cot}(胃) = \frac{1}{\text{tan}(胃)}\)
  • Secant (Sec): \(\text{sec}(胃) = \frac{1}{\text{cos}(胃)}\)
  • Cosecant (Csc): \(\text{csc}(胃) = \frac{1}{\text{sin}(胃)}\)

Understanding these functions is critical for solving many real-world problems, from calculating heights of objects using angles of elevation to analyzing sound waves and electrical currents.

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Most popular questions from this chapter

Use the given information to find ( \(a\) ) \(\sin (s+t),(b) \tan (s+t),\) and \((c)\) the quadrant of \(s+t .\) $$\cos s=\frac{3}{5} \text { and } \sin t=\frac{5}{13}, s \text { and } t \text { in quadrant } \mathbf{I}$$

Solve each problem. When a musical instrument creates a tone of \(110 \mathrm{Hz}\). it also creates tones at \(220,330,440,550,660, \ldots\) Hz. A small speaker cannot reproduce the \(110-\mathrm{Hz}\) vibration but it can reproduce the higher frequencies, which are the upper harmonics. The low tones can still be heard because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is \(110 \mathrm{Hz}\), and this difference tone will be heard by a listener. (Source: Benade, A.. Fundamentals of Musical Acoustics, Dover Publications.) (a) We can model this phenomenon using a graphing calculator. In the window \([0,0.03]\) by \([-1,1],\) graph the upper harmonics represented by the pressure $$ P=\frac{1}{2} \sin [2 \pi(220) t]+\frac{1}{3} \sin [2 \pi(330) t]+\frac{1}{4} \sin [2 \pi(440) t] $$ (b) Estimate all \(t\) -coordinates where \(P\) is maximum. (c) What does a person hear in addition to the frequencies of \(220,330,\) and \(440 \mathrm{Hz} ?\) (d) Graph the pressure produced by a speaker that can vibrate at \(110 \mathrm{Hz}\) and above.

Verify that each trigonometric equation is an identity. $$\sin ^{2} x(1+\cot x)+\cos ^{2} x(1-\tan x)+\cot ^{2} x=\csc ^{2} x$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin \frac{\theta}{2}=1$$

Verify that each trigonometric equation is an identity. $$\frac{1-\cos x}{1+\cos x}=\csc ^{2} x-2 \csc x \cot x+\cot ^{2} x$$
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