/*! 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 106 Evaluate the function for the gi... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 106

Evaluate the function for the given values of \(t\). Values of \(t\) (a) \(t=3\) (a) \(t=2\) (a) \(t=0.3\) (a) \(t=0.1\) (b) \(t=\pi\) (b) \(t=0\) (b) \(t=-\pi / 4\) (b) \(t=-2 \pi / 3\) Trigonometric Function $$f(t)=\cos t$$

Short Answer

Expert verified
The values of \( f(t) = \cos t \) at the provided points are: \( f(3) = -0.9900 \), \( f(2) = -0.4161 \), \( f(0.3) = 0.9553 \), \( f(0.1) = 0.9950 \), \( f(\pi) = -1 \), \( f(0) = 1 \), \( f(-\pi/4) = 0.7071 \), \( f(-2\pi/3) = -0.5000 \)

Step by step solution

01

Evaluate \( f(t) = \cos t \) for \( t = 3 \)

Replace \( t \) with 3 in the function. The value will be \( f(3) = \cos(3) \). Using a calculator the value will be approximately equal to -0.9900.
02

Evaluate \( f(t) = \cos t \) for \( t = 2 \)

Replace \( t \) with 2 in the function. The value will be \( f(2) = \cos(2) \). Using a calculator the value will be approximately equal to -0.4161.
03

Evaluate \( f(t) = \cos t \) for \( t = 0.3 \)

Replace \( t \) with 0.3 in the function. The value will be \( f(0.3) = \cos(0.3) \). Using a calculator, this will be approximately equal to 0.9553.
04

Evaluate \( f(t) = \cos t \) for \( t = 0.1 \)

Replace \( t \) with 0.1 in the function. The value will be \( f(0.1) = \cos(0.1) \). Using a calculator, this will be approximately equal to 0.9950.
05

Evaluate \( f(t) = \cos t \) for \( t = \pi \)

Replace \( t \) with \( \pi \) in the function. The value will be \( f(\pi) = \cos(\pi) \). We know that \(\cos(\pi) = -1\)
06

Evaluate \( f(t) = \cos t \) for \( t = 0 \)

Replace \( t \) with 0 in the function. The value will be \( f(0) = \cos(0) \). We know that \(\cos(0) = 1 \)
07

Evaluate \( f(t) = \cos t \) for \( t = -\pi / 4 \)

Replace \( t \) with \( -\pi/4 \) in the function. The value will be \( f(-\pi/4) = \cos(-\pi/4) \). Using a calculator, this will be approximately equal to 0.7071.
08

Evaluate \( f(t) =\cos t \) for \( t = -2\pi/3 \)

Replace \( t \) with \( -2\pi/3 \) in the function. The value will be \( f(-2\pi/3) = \cos(-2\pi/3) \). Using a calculator, this will be approximately equal to -0.5000.

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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 \( \cos t \), is one of the fundamental trigonometric functions used to link angles to the ratios of sides in right-angled triangles. But its use extends beyond just triangles, describing wave patterns like sound and light. The cosine function provides the horizontal coordinate of a unit circle corresponding to a particular angle \( t \). This results in values that range between \(-1\) and \(1\).
  • When \( t = 0 \), the cosine is \(1\).
  • At \( t = \pi \), the cosine is \(-1\).
  • For angles \( t \) like \( \pi/2 \) and \( 3\pi/2 \), the cosine takes the value \(0\).
Cosine is an even function, meaning that \( \cos(-t) = \cos(t) \). This symmetry makes it easier to determine values for negative angles.
Evaluation of Functions
Evaluating a function involves finding the value of \( f(t) \) for given inputs. For \( f(t) = \cos t \), we substitute \( t \) with the given values and use either trigonometric knowledge or a calculator to find \( \cos(t) \).
  • For example, \( f(3) = \cos(3) \), which can be approximately calculated with a calculator.
  • Standard angles like \( \pi \) or \( 0 \) have known cosine values: \(\cos(\pi) = -1 \) and \( \cos(0) = 1 \).
When using a calculator, ensure it's set to the correct mode (radians or degrees) depending on your input angle \( t \), as this affects the outcome of your evaluation.
Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It's a crucial tool in trigonometry for understanding angle and trigonometric function relationships.
  • The cosine of an angle \( t \) in the unit circle corresponds to the x-coordinate of the point on the circle's circumference.
  • This visualization helps bind the theoretical understanding of trigonometric functions to geometric representations.
The unit circle aids in converting between angle measures and finding precise trigonometric function values for angles, bridging algebraic and geometric aspects of trigonometry.
Angle Measurement
Angles in trigonometry can be measured in degrees or radians. For the cosine function, radians are commonly used because they seamlessly integrate with calculus and other branches of mathematics.
  • One complete revolution around the circle is \( 360\) degrees or \( 2\pi \) radians.
  • Understanding radians helps in simplifying the measurement of angles, especially those not marked by common degree values, like \( \pi/4 \) or \( -2\pi/3 \).
It's essential to become familiar with converting between degrees and radians: \( 180 \) degrees equals \( \pi \) radians, providing a standardized conversion factor for calculations.

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Most popular questions from this chapter

A 20 -meter line is used to tether a helium-filled balloon. Because of a breeze, the line makes an angle of approximately \(85^{\circ}\) with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the balloon? (d) The breeze becomes stronger, and the angle the balloon makes with the ground decreases. How does this affect your triangle from part (a)? (e) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures \(\theta\). $$\begin{array}{|l|l|l|l|l|}\hline \text { Angle, } \theta & 80^{\circ} & 70^{\circ} & 60^{\circ} & 50^{\circ} \\\\\hline \text { Height } & & & & \\\\\hline\end{array}$$ $$\begin{array}{|l|l|l|l|l|}\hline \text { Angle, } \theta & 40^{\circ} & 30^{\circ} & 20^{\circ} & 10^{\circ} \\\\\hline \text { Height } & & & & \\\\\hline\end{array}$$ (f) As the angle the balloon makes with the ground approaches \(0^{\circ},\) how does this affect the height of the balloon? Draw a right triangle to explain your reasoning.

Solve the equation. Round your answer to three decimal places, if necessary. $$x^{2}-2 x-5=0$$

In calculus, it is shown that the area of the region bounded by the graphs of \(y=0, y=1 /\left(x^{2}+1\right), x=a,\) and \(x=b\) is given by Arca \(=\arctan b-\arctan a\) (see figure). Find the area for each value of \(a\) and \(b\) (a) \(a=0, b=1\) (b) \(a=-1, b=1\) (c) \(a=0, b=3\) (d) \(a=-1, b=3\)

Solve the equation. Round your answer to three decimal places, if necessary. $$2 x^{2}+x-4=0$$

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