Problem 53

Question

From the information given, find the quadrant in which the terminal point determined by $t$ lies. $\csc t > 0$ and $\sec t < 0$

Step-by-Step Solution

Verified

Answer

The terminal point lies in the second quadrant.

1Step 1: Understanding Trig Functions

First, let's understand what the csc and sec functions represent with respect to the unit circle. The cosecant function, $\csc t$, is the reciprocal of $\sin t$, meaning $\csc t = \frac{1}{\sin t}$. A positive $\csc t$ implies a positive $\sin t$. The secant function, $\sec t$, is the reciprocal of $\cos t$, meaning $\sec t = \frac{1}{\cos t}$. A negative $\sec t$ implies a negative $\cos t$.

2Step 2: Analyzing Positive Sine

Given $\csc t > 0$, we deduce that $\sin t > 0$. In the unit circle, $\sin t$ is positive in the first quadrant (corresponding angle $0$ to $\frac{\pi}{2}$) and the second quadrant ($\frac{\pi}{2}$ to $\pi$).

3Step 3: Analyzing Negative Cosine

Given $\sec t < 0$, we deduce that $\cos t < 0$. In the unit circle, $\cos t$ is negative in the second quadrant ($\frac{\pi}{2}$ to $\pi$) and the third quadrant ($\pi$ to $\frac{3\pi}{2}$).

4Step 4: Determining the Common Quadrant

Since $\csc t > 0$ implies the terminal point could be in either the first or second quadrant, and $\sec t < 0$ indicates it could be in the second or third quadrant, the only quadrant common to both is the second quadrant. Therefore, the terminal point determined by $t$ lies in the second quadrant.

Key Concepts

Understanding Trigonometric FunctionsIdentifying Quadrants on the Unit CircleExploring Cosecant and Secant

Understanding Trigonometric Functions

Trigonometric functions are fundamental to understanding angles and their relationships in a unit circle. There are six key trigonometric functions: sine ($\sin\theta $), cosine ($\cos\theta $), tangent ($\tan\theta $), cosecant ($\csc\theta $), secant ($\sec\theta $), and cotangent ($\cot\theta $).

Sine ($\sin\theta$) represents the y-coordinate on the unit circle.
Cosine ($\cos\theta$) represents the x-coordinate on the unit circle.
Tangent ($\tan\theta$) is the ratio of sine to cosine.
Cosecant ($\csc\theta$) is the reciprocal of sine: $\csc\theta$ = $\frac{1}{\sin\theta}$.
Secant ($\sec\theta$) is the reciprocal of cosine: $\sec\theta$ = $\frac{1}{\cos\theta}$.
Cotangent ($\cot\theta$) is the reciprocal of tangent.

Understanding these functions is essential when working with angles, especially when determining their signs in different quadrants. The unit circle helps visualize these functions as it maps the functions to points on a circle with radius one, centered at the origin (0,0).

Identifying Quadrants on the Unit Circle

The unit circle is divided into four quadrants, with each quadrant associated with specific ranges for angles.

First Quadrant: Angles range from 0 to $\frac{\pi}{2}$ radians. Here, both sine and cosine are positive.
Second Quadrant: Angles range from $\frac{\pi}{2}$ to $\pi$ radians. Sine is positive, but cosine is negative.
Third Quadrant: Angles range from $\pi$ to $\frac{3\pi}{2}$ radians. In this region, both sine and cosine are negative.
Fourth Quadrant: Angles range from $\frac{3\pi}{2}$ to $2\pi$ radians. Here, sine is negative but cosine is positive.

Understanding which quadrant an angle resides in helps determine the signs of trigonometric functions. For the given problem, knowing that cosecant is positive and secant is negative helps you find the correct quadrant. Positive sine corresponds to the first and second quadrants, whereas a negative cosine corresponds to the second and third quadrants. Therefore, the intersection of these conditions occurs in the second quadrant.

Exploring Cosecant and Secant

Cosecant and secant are less commonly used trigonometric functions, but they hold significant importance because they are the reciprocals of sine and cosine, respectively.

Cosecant Function

Since cosecant ($\csc t$) is the reciprocal of sine, it is undefined when the sine value is zero (e.g., at angles 0, $\pi$, 2$\pi$, etc). A positive $\csc t$ implies that the sine function is also positive, which occurs in the first and second quadrants.

Secant Function

Secant ($\sec t$) is the reciprocal of cosine. It becomes undefined when cosine is zero, hitting these points at angles $\frac{\pi}{2}$ and $\frac{3\pi}{2}$. A negative $\sec t$ suggests the cosine value is negative, which happens in the second and third quadrants.
To correctly identify the quadrant that matches $\csc t > 0$ and $\sec t < 0$, one needs to find where both conditions are satisfied simultaneously. With $\sin t$ positive and $\cos t$ negative exclusively in the second quadrant, this is the area where the terminal side of angle $t$ resides.

Problem 52

Problem 53

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Problem 52

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Problem 53

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Problem 54

From the information given, find the quadrant in which the terminal point determined by $t$ lies. \(\cos t

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