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Chapter 4: Problem 26

Evaluate (if possible) the six trigonometric functions at the real number. $$t=7 \pi / 4$$

Short Answer

Expert verified
The values of the six trigonometric functions at \(t = 7\pi / 4\) are: \(\sin(t) = -\sqrt{2}/2\), \(\cos(t) = \sqrt{2}/2\), \(\tan(t) = -1\), \(\csc(t) = -\sqrt{2}\), \(\sec(t) = \sqrt{2}\) and \( \cot(t) = -1\).

Step by step solution

01

Place the Value on the Unit Circle

The value \(t = 7 \pi / 4\) is given in radian measure. If related to the unit circle, this value refers to a point counterclockwise from the positive x-axis. This is got by taking \(7/4\) revolutions around the unit circle from 0.
02

Establish Coordinates of the Point

The point on the unit circle that corresponds to the angle \(7 \pi / 4\) radians is the same as for \(\pi / 4\), or 45 degrees, due to symmetry. The point corresponding to 45 degree or \(\pi / 4\) is \((\sqrt{2}/2, \sqrt{2}/2)\). The point corresponding to \(7\pi / 4\) is thus \((\sqrt{2}/2, -\sqrt{2}/2)\) because it is positioned in the 4th quadrant where \(x\) is positive and \(y\) is negative.
03

Evaluate Trigonometric Functions

Now evaluate each of the six trigonometric functions at \(t=7\pi / 4\). By considering the coordinates \(x = \sqrt{2}/2\) and \(y = -\sqrt{2}/2\), we evaluate as follows: \(\sin(t) = y = -\sqrt{2}/2, \cos(t) = x = \sqrt{2}/2, \tan(t) = y/x = -1, \csc(t) = 1/y = -\sqrt{2}, \sec(t) = 1/x = \sqrt{2} \) and \( \cot(t) = 1/(y/x) = -1\).

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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, serving as a bridge between geometry and trigonometric functions. Essentially, it is a circle with a radius of one, perfectly centered at the origin of the coordinate plane
This simple geometric construct enables us to evaluate trigonometric functions at any given angle.
  • In the unit circle, the coordinates of a point corresponding to an angle \(\theta\) from the positive x-axis are \((\cos\theta, \sin\theta)\).
  • The x-coordinate gives the value of \(\cos\theta\), while the y-coordinate reveals \(\sin\theta\).
  • The circle is divided into four quadrants, each exhibiting a specific sign pattern for trigonometric functions based on the axes.
For the angle \(7\pi/4\), we start from the positive x-axis and move counterclockwise. The point lands in the fourth quadrant where \(x\) remains positive and \(y\) turns negative. Recognizing the quadrant for any angle helps in determining the sign of the trigonometric values.
Radian Measure
Radian measure is a way of measuring angles based on the radius of a circle. Unlike degrees, which are somewhat arbitrary, radians have a more natural geometric interpretation.
One radian is the angle made when the arc length is equal to the radius of the circle. The entire circle amounts to \(2\pi\) radians (equivalent to 360 degrees), serving as the complete rotation.
  • An angle in radians can be found by dividing the length of the arc by the radius.
  • For practical purposes, angles are often expressed in terms of \(\pi\), making trigonometric calculations simpler.
  • Radian measures are crucial when dealing with periodic functions or phenomena as they maintain continuity.
In our example, \(t = 7\pi/4\) radians represents a rotation that also corresponds to an angle in the fourth quadrant. By understanding how radians work, it becomes easier to imagine and locate the angle on the unit circle.
Trigonometric Identities
Trigonometric identities are equations that hold for all permissible values of the variable where the functions are defined. These identities simplify expressions and solve trigonometric equations.
Key identities often include:
  • Pythagorean identities such as \(\sin^2\theta + \cos^2\theta = 1\).
  • Reciprocal identities like \(\csc\theta = 1/\sin\theta\) and \(\sec\theta = 1/\cos\theta\).
  • Quotient identities, where \(\tan\theta = \sin\theta/\cos\theta\).
Using the point \((\sqrt{2}/2, -\sqrt{2}/2)\) for \(7\pi/4\) on the unit circle, we derive the trigonometric functions:
  • \(\sin(7\pi/4) = -\sqrt{2}/2\)
  • \(\cos(7\pi/4) = \sqrt{2}/2\)
  • \(\tan(7\pi/4) = -1\)
  • \(\csc(7\pi/4) = -\sqrt{2}\)
  • \(\sec(7\pi/4) = \sqrt{2}\)
  • \(\cot(7\pi/4) = -1\)
Understanding these identities helps in converting between functions and simplifying calculations in various applications.

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Most popular questions from this chapter

A ship leaves port at noon and has a bearing of \(\mathrm{S} 29^{\circ} \mathrm{W}\). The ship sails at 20 knots. (a) How many nautical miles south and how many nautical miles west will the ship have traveled by 6: 00 P.M.? (b) At 6: 00 P.M., the ship changes course to due west. Find the ship's bearing and distance from the port of departure at 7: 00 P.M.

The bearing \(\mathrm{N} 24^{\circ} \mathrm{E}\) means 24 degrees north of east.

Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals \([-\pi / 2,0)\) and \((0, \pi / 2],\) and sketch the graph of the inverse trigonometric function.

The Johnstown Inclined Plane in Pennsylvania is one of the longest and steepest hoists in the world. The railway cars travel a distance of 896.5 feet at an angle of approximately \(35.4^{\circ},\) rising to a height of 1693.5 feet above sea level. (a) Find the vertical rise of the inclined plane. (b) Find the elevation of the lower end of the inclined plane. (c) The cars move up the mountain at a rate of 300 feet per minute. Find the rate at which they rise vertically.

Use a graphing utility to graph the functions \(f(x)=\sqrt{x}\) and \(g(x)=6\) arctan \(x .\) For \(x>0,\) it appears that \(g>f .\) Explain why you know that there exists a positive real number \(a\) such that \(ga .\) Approximate the number \(a.\)
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