Problem 17 - Algebra and Trigonometry

Question

let \(\theta\) be an angle in standard position. Name the quadrant in which \(\theta\) lies. $$ \sin \theta>0, \quad \cos \theta>0 $$

Step-by-Step Solution

Answer

The angle \(θ\) lies in Quadrant I.

Step 1: Understanding Trigonometric Function Signs

In the unit circle, the \(\sin θ\) represents the y-coordinate and the \(\cos θ\) the x-coordinate. The sign of \(\sin θ\) and \(\cos θ\) can help us to figure out the quadrant: - Quadrant I: \(sin θ > 0\) and \(cos θ > 0\)- Quadrant II: \(sin θ > 0\) and \(cos θ < 0\)- Quadrant III: \(sin θ < 0\) and \(cos θ < 0\)- Quadrant IV: \(sin θ < 0\) and \(cos θ > 0\)

Step 2: Identifying the Quadrant

Given that \(\sin θ > 0\) and \(\cos θ > 0\), this corresponds to the conditions for Quadrant I.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate plane. This simple setup provides powerful tools for understanding trigonometric functions.
In the unit circle, any point \(x, y\) on the circle represents the cosine and sine of an angle \(\theta\) that's measured from the positive x-axis.

• The x-coordinate corresponds to \(\cos \theta\).
• The y-coordinate represents \(\sin \theta\).

By looking at these coordinates on the unit circle, you can determine the values of trigonometric functions for any given angle. The unit circle helps visualize how these functions behave as the angle changes, making it indispensable for solving trigonometry problems.

Angle in Standard Position

When we refer to an angle in standard position, we mean an angle that is positioned in a certain way on the coordinate plane.
This setup has specific characteristics:

• The vertex is at the origin \( (0,0) \).
• The initial side lies along the positive x-axis.
• The angle opens in a counter-clockwise direction.

This arrangement makes it easier to evaluate trigonometric functions since the angle's location and size are consistent. By using standard position, interpreting angles becomes more straightforward, especially when working with the unit circle to find sine and cosine values.

Quadrants

The coordinate plane is divided into four quadrants, each offering unique characteristics for trigonometric functions:

• **Quadrant I**: Both \(\sin \theta\) and \(\cos \theta\) are positive.
• **Quadrant II**: \(\sin \theta > 0\), \(\cos \theta < 0\).
• **Quadrant III**: Both \(\sin \theta\) and \(\cos \theta\) are negative.
• **Quadrant IV**: \(\sin \theta < 0\), \(\cos \theta > 0\).

This division plays a crucial role in determining the sign of trigonometric functions and helps identify an angle's quadrant just by knowing the signs of its sine and cosine. In the given exercise, knowing that both \(\sin \theta > 0\) and \(\cos \theta > 0\) indicates that the angle lies in Quadrant I.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved angles. They provide tools for simplifying expressions and solving equations.
Key trigonometric identities include:

• **Pythagorean Identity**: \(\sin^2 \theta + \cos^2 \theta = 1\)
• **Reciprocal Identities**: Relationships involving \(\tan\), \(\sec\), \(\cot\), and \(\csc\).

These identities allow us to express one trigonometric function in terms of another, facilitating easier calculations. They are particularly useful when dealing with complex trigonometric expressions or when verifying solutions. Understanding and employing these identities aid significantly in solving and understanding the breadth of trigonometric problems.