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Chapter 5: Problem 128

Find the five remaining trigonometric finction values for each angle. \(\cos \theta=-\frac{3}{3},\) and \(\theta\) is in quadrant III.

Short Answer

Expert verified
The five remaining trigonometric function values are: \(\sin \theta = 0\), \(\tan \theta = 0\), \(\csc \theta\) is undefined, \(\sec \theta = -1\), \(\cot \theta\) is undefined.

Step by step solution

01

Simplify and identify cosine

First, simplify \(\text{\(\frac{3}{3}\)}\) to get -1. This indicates that \(\cos \theta = -1\).
02

Determine the sine

In quadrant III, both sine and cosine are negative. Using \(\cos^2 \theta + \sin^2 \theta = 1\), and since \(\cos \theta = -1\): \(\sin^2 \theta = 1 - (-1)^2 = 1 - 1 = 0\). Thus, \(\sin \theta = 0\).
03

Calculate tangent

Tangent is the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{0}{-1} = 0\).
04

Find cosecant

Cosecant is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta}\). Since \(\sin \theta = 0\), \(\csc \theta\) is undefined.
05

Find secant

Secant is the reciprocal of cosine: \(\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-1} = -1\).
06

Find cotangent

Cotangent is the reciprocal of tangent: \(\cot \theta = \frac{1}{\tan \theta} = \frac{1}{0} = \text{undefined}\).

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Key Concepts

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

Cosine
The cosine function, denoted as \(\text{cos} \theta\), describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. For an angle \(\theta\) in standard position, cosine can also be understood on the unit circle as the x-coordinate of the point where the terminal side intersects the circle.
In this problem, \(\theta\) is in quadrant III, meaning both sine and cosine are negative here. Given \(\text{cos} \theta = -1\), we are dealing with the point \((-1, 0)\) on the unit circle. This direct information simplifies a lot of subsequent calculations.
Sine
The sine function, denoted as \(\text{sin} \theta\), represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. On the unit circle, it corresponds to the y-coordinate of the intersection point.
In quadrant III, sine is negative, and using the Pythagorean identity \(\text{cos}^2 \theta + \text{sin}^2 \theta = 1\): \(\text{sin}^2 \theta = 1 - (-1)^2 = 0\). Thus, \(\text{sin} \theta = 0\).
Tangent
Tangent is the ratio of the sine to the cosine of an angle, expressed as \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\). Tangent reflects the slope of the line created by the angle in standard position.
For the given problem, where \(\text{sin} \theta = 0\) and \(\text{cos} \theta = -1\), \(\text{tan} \theta = \frac{0}{-1} = 0\). Because both sine and cosine are involved, \(\text{tan} \theta\) also inherits the sign based on the quadrant.
Cosecant
Cosecant, denoted as \(\text{csc} \theta\), is the reciprocal of sine. It is defined as \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\).
When sine is zero, as in our problem, cosecant becomes undefined. It's important to remember that functions involving division by zero are undefined in mathematics, which results in \(\text{csc} \theta\) having no value here. This underscores the importance of never dividing by zero.
Secant
Secant, denoted as \(\text{sec} \theta\), is the reciprocal of cosine. Its definition is \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\).
For our given angle where \(\text{cos} \theta = -1\), \(\text{sec} \theta = \frac{1}{-1} = -1\).
In this sense, secant and cosine share a direct relationship that often simplifies solving trigonometric identities.
Cotangent
Cotangent is the reciprocal of tangent, written as \(\text{cot} \theta = \frac{1}{\text{tan} \theta}\).
Because the tangent of our specific angle is zero, cotangent becomes undefined, as \(\text{tan} \theta\) appears in the denominator. This is another case where dividing by zero leads to an undefined value.
\(\text{cot} \theta\) is thus a crucial reciprocal function that can be understood better by first calculating the tangent.

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Most popular questions from this chapter

In one area, the lowest angle of elevation of the sun in winter is \(23^{\circ} 20^{\prime} .\) Find the minimum distance \(x\) that a plant needing full sun can be placed from a fence 4.65 ft high. (IMAGES CANNOT COPY)

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Find the five remaining trigonometric finction values for each angle. \(\sin \theta=\frac{\sqrt{5}}{7},\) and \(\theta\) is in quadrant I.

Suppose that \(-90^{\circ}<\theta<90^{\circ}\). Find the sign of each finction value. $$\cos (-\theta)$$

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. $$1005^{\circ}$$
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