Question 10 Given That Cos(0) = = 10 Provide Your Answer Below: Sin (20) = And Is In Quadrant III, What
Understanding trigonometric functions and their values is fundamental in mathematics, especially in the study of angles and their relationships on the unit circle. The question presented appears to contain some ambiguities, but it hinges on key concepts such as the cosine and sine functions, their values at specific angles, and the implications of angle placement within different quadrants. In this comprehensive guide, we will explore the properties of cosine and sine functions, interpret the given information, clarify common misconceptions, and provide a step-by-step approach to solving similar trigonometric problems.
---
Interpreting the Question: Clarifications and Assumptions
Before delving into calculations, it is crucial to interpret the question accurately. The question states:
"Question 10 Given That Cos(0) = = 10 Provide Your Answer Below: Sin (20) = And Is In Quadrant III, What"
Several points need clarification:
- Cos(0) = 10: Typically, the cosine of an angle in radians or degrees ranges between -1 and 1. A value of 10 suggests either a typographical error or a misinterpretation. Likely, the intended notation is cos(θ) = 10, which is impossible within a real-number context for a standard cosine function. Alternatively, it might be a typo, and the intended value could be cos(0) = 1.
- Sin (20): Usually, this refers to sin(20°) or sin(20 radians). Since degrees are more common in basic trigonometry, we will assume 20° unless specified otherwise.
- And Is In Quadrant III: This indicates that the angle in question (possibly 20°) is located in the third quadrant, where both sine and cosine are negative.
Given these observations, it's reasonable to reframe the question as follows:
> Given that cos(θ) = 10 (which is impossible unless considering complex numbers), or perhaps, more realistically, the problem intends to say that some angle's cosine is a certain value, and the angle in question (say, 20°) lies in Quadrant III.
Alternatively, perhaps the question is asking:
- If an angle in Quadrant III has certain properties, what is the sine value at 20°, given that the cosine value at some point is known?
Due to the ambiguity, we will proceed with the most reasonable assumptions:
- The cosine value at the angle is cos(θ) = -0.5 (or some valid value), consistent with Quadrant III where cosine and sine are both negative.
- The angle 20° is in Quadrant III, which is impossible because 20° is in Quadrant I, but perhaps the question refers to an angle that is "reference angle" in Quadrant III.
- The main goal is to find sin(20°) given that the angle is in Quadrant III, where both sine and cosine are negative.
---
Fundamentals of Trigonometric Functions and Quadrants
To understand the problem thoroughly, let's revisit some core concepts:
1. The Unit Circle and Trigonometric Ratios
The unit circle is a fundamental tool in trigonometry, representing all possible angles and their sine and cosine values:
- Cosine of an angle corresponds to the x-coordinate of the point on the unit circle.
- Sine corresponds to the y-coordinate.
The values of sine and cosine vary depending on the angle's position in the four quadrants:
| Quadrant | Range of Angles | Sign of Sine | Sign of Cosine |
|------------|-----------------|--------------|----------------|
| I | 0° to 90° | Positive | Positive |
| II | 90° to 180° | Positive | Negative |
| III | 180° to 270° | Negative | Negative |
| IV | 270° to 360° | Negative | Positive |
Note: In Quadrant III, both sine and cosine are negative.
2. Basic Trigonometric Identities
Some essential identities include:
- Pythagorean Identity:
\sin^2 \theta + \cos^2 \theta = 1
\]
- Sign of Sine and Cosine in Quadrants:
- Quadrant I: \(\sin \theta > 0,\ \cos \theta > 0\)
- Quadrant II: \(\sin \theta > 0,\ \cos \theta < 0\)
- Quadrant III: \(\sin \theta < 0,\ \cos \theta < 0\)
- Quadrant IV: \(\sin \theta < 0,\ \cos \theta > 0\)
---
Analyzing the Given Data and Constraints
Given the ambiguity, let's consider two possible interpretations:
Scenario 1: The Value of Cosine is Corrected to a Valid Range
Suppose the question intended to say:
> "Given that \(\cos \theta = -\frac{1}{2}\) and \(\theta\) is in Quadrant III, find \(\sin \theta\)."
In this case:
- Since \(\cos \theta = -\frac{1}{2}\),
- And \(\theta\) is in Quadrant III (where both sine and cosine are negative),
we can find \(\sin \theta\) using the Pythagorean identity.
Scenario 2: The Angle is 20°, and the question asks for the sine value in Quadrant III
Given that 20° is in Quadrant I, but the question states "And Is In Quadrant III", perhaps it's referring to a reference angle or an angle coterminal or related to 20°, but located in Quadrant III.
---
Step-by-Step Solution Approach
Based on the most logical correction—Scenario 1—we will proceed to find \(\sin \theta\) given \(\cos \theta = -\frac{1}{2}\) and \(\theta\) in Quadrant III.
---
Step 1: Confirm the Quadrant
- Since \(\theta\) is in Quadrant III, both sine and cosine are negative.
- \(\cos \theta = -\frac{1}{2}\) is consistent with Quadrant III.
Step 2: Use the Pythagorean Identity to Find \(\sin \theta\)
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Plug in \(\cos \theta = -\frac{1}{2}\):
\[
\sin^2 \theta + \left(-\frac{1}{2}\right)^2 = 1
\]
\[
\sin^2 \theta + \frac{1}{4} = 1
\]
\[
\sin^2 \theta = 1 - \frac{1}{4} = \frac{3}{4}
\]
\[
\sin \theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}
\]
Since \(\theta\) is in Quadrant III, where sine is negative:
\[
\boxed{
\sin \theta = - \frac{\sqrt{3}}{2}
}
\]
---
Step 3: Final Answer and Interpretation
Given the corrected assumption, the sine of the angle in Quadrant III with \(\cos \theta = -\frac{1}{2}\) is
\[
\boxed{
\sin \theta = - \frac{\sqrt{3}}{2}
}
\]
---
Additional Related Concepts
- Finding Reference Angles
\[
\alpha = \arccos \left( \frac{1}{2} \right) = 60^\circ
\]
Since \(\theta\) is in Quadrant III, its measure is:
\[
\theta = 180^\circ + 60^\circ = 240^\circ
\]
At this angle:
\[
\sin 240^\circ = - \frac{\sqrt{3}}{2}
\]
which confirms our earlier calculation.
- Common Trigonometric Values
| Angle (degrees) | \(\sin \theta\) | \(\cos \theta\) |
|-------------------|----------------|----------------|
| 30° | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) |
| 60° | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) |
| 120° | \(\frac{\sqrt{3}}{2}\) | \(-\frac{1}{2}\) |
| 180° | 0 | -1 |
| 210
