Problem 19
Question
State the quadrant in which \(\theta\) lies. $$ \sin \theta>0 \text { and } \cos \theta>0 $$
Step-by-Step Solution
Verified Answer
The angle \(\theta\) lies in the first quadrant.
1Step 1: Identify Quadrant for Sin(θ)>0
Understand that sine is positive in the first and second quadrant.
2Step 2: Identify Quadrant for Cos(θ)>0
Recognize that cosine is positive in the first and fourth quadrant.
3Step 3: Find Common Quadrant
Identify the common quadrant where both \(\sin \theta\) and \(\cos \theta\) are positive which is the first quadrant.
Key Concepts
Sine and Cosine in TrigonometryUnderstanding the Quadrant SystemTrigonometry in the Unit Circle
Sine and Cosine in Trigonometry
To thoroughly grasp the relationship between sine and cosine values, we must first consider their definitions within the context of the unit circle. The sine of an angle \theta, denoted by \( \sin \theta \), corresponds to the y-coordinate of the point where the angle's terminal side intersects the unit circle. Conversely, the cosine of \theta, or \( \cos \theta \), is the x-coordinate of this intersection point.
When given a condition like \( \sin \theta > 0 \) and \( \cos \theta > 0 \), as in our exercise, we're dealing with angles that produce positive y-coordinates and positive x-coordinates on the unit circle. This directly implies that these angles correspond to a section of the circle where both the x and y values are positive - essentially, the arc where both functions’ values will be above zero. The immediate implication is that the angle \( \theta \) must be located within the quadrantal space that supports these positive values.
When given a condition like \( \sin \theta > 0 \) and \( \cos \theta > 0 \), as in our exercise, we're dealing with angles that produce positive y-coordinates and positive x-coordinates on the unit circle. This directly implies that these angles correspond to a section of the circle where both the x and y values are positive - essentially, the arc where both functions’ values will be above zero. The immediate implication is that the angle \( \theta \) must be located within the quadrantal space that supports these positive values.
Understanding the Quadrant System
The quadrant system is a crucial concept for understanding the signs of trigonometric functions. The Cartesian plane is divided into four quadrants by the x-axis (horizontal) and the y-axis (vertical), with each quadrant representing a range of angle measures:
The sign of sine and cosine changes depending on the quadrant in which an angle terminates. In the first quadrant, both sine and cosine are positive. In the second quadrant, sine is positive while cosine is negative, and so on.
Understanding these sign changes is essential because it helps us determine where an angle lies based on the signs of its trigonometric functions, as shown in the provided exercise.
- The first quadrant contains angles from 0 to 90 degrees.
- The second quadrant ranges from 90 to 180 degrees.
- The third quadrant covers angles from 180 to 270 degrees.
- The fourth quadrant includes angles from 270 to 360 degrees.
The sign of sine and cosine changes depending on the quadrant in which an angle terminates. In the first quadrant, both sine and cosine are positive. In the second quadrant, sine is positive while cosine is negative, and so on.
Understanding these sign changes is essential because it helps us determine where an angle lies based on the signs of its trigonometric functions, as shown in the provided exercise.
Trigonometry in the Unit Circle
When discussing trigonometry in the unit circle, we focus on angles whose vertex is at the origin and whose initial side lies along the positive x-axis. The radius of the unit circle is one, which simplifies calculations and visualizations of trigonometric functions.
Each point along the circumference of the unit circle can be represented by \( (\cos \theta, \sin \theta) \), linking the geometric and algebraic aspects of trigonometry. The unit circle approach provides a comprehensive way to see how sine and cosine values behave across different angles - a concept crucial in solving problems like the one given in the textbook exercise.
With the knowledge that sine relates to the vertical coordinate and cosine to the horizontal, it becomes evident that for \( \sin \theta > 0 \) and \( \cos \theta > 0 \) the terminal side of \( \theta \) must lie in the first quadrant. This method is not only useful for single-variable problems, but also for understanding more complex trigonometric concepts involving other functions and the relationships between them.
Each point along the circumference of the unit circle can be represented by \( (\cos \theta, \sin \theta) \), linking the geometric and algebraic aspects of trigonometry. The unit circle approach provides a comprehensive way to see how sine and cosine values behave across different angles - a concept crucial in solving problems like the one given in the textbook exercise.
With the knowledge that sine relates to the vertical coordinate and cosine to the horizontal, it becomes evident that for \( \sin \theta > 0 \) and \( \cos \theta > 0 \) the terminal side of \( \theta \) must lie in the first quadrant. This method is not only useful for single-variable problems, but also for understanding more complex trigonometric concepts involving other functions and the relationships between them.
Other exercises in this chapter
Problem 19
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