Problem 23 - Algebra 2 and Trigonometry

Question

In \(21-26,\) if \(\theta\) is the measure of \(\angle A O B,\) an angle in standard position, name the quadrant in which the terminal side of \(\angle A O B\) lies. $$ \sin \theta<0, \cos \theta<0 $$

Step-by-Step Solution

Answer

The terminal side of the angle lies in Quadrant III.

Step 1: Understanding the Angles

In standard position, the angle \(\theta\) is measured from the positive x-axis. Quadrants are ordered counter-clockwise from the positive x-axis: Quadrant I (top right), Quadrant II (top left), Quadrant III (bottom left), and Quadrant IV (bottom right).

Step 2: Analyzing the Given Conditions

We're provided with two conditions: \( \sin \theta < 0 \) and \( \cos \theta < 0 \). A sine value less than 0 indicates that the angle lies in either Quadrant III or IV, since the sine represents the y-coordinate, which is negative in these quadrants. Similarly, a cosine value less than 0 indicates the angle is in either Quadrant II or III, since the cosine represents the x-coordinate, which is negative in these quadrants.

Step 3: Identifying the Common Quadrant

To satisfy both conditions, the terminal side of the angle must lie in the quadrant where both the sine and cosine are negative. The only quadrant where both conditions are true is Quadrant III, where both x and y values are negative, making both \( \sin \theta\) and \( \cos \theta\) negative.

Key Concepts

Angle in Standard Position

In trigonometry, an angle is said to be in standard position when its vertex is located at the origin of a coordinate plane. The initial side of the angle coincides with the positive x-axis. From this starting point, the terminal side can rotate. This rotation can either be counter-clockwise, which indicates a positive angle, or clockwise, which indicates a negative angle.

Angles in standard position are categorized based on the quadrant where their terminal side lands. The quadrants in a coordinate plane are numbered counter-clockwise starting from the positive x-axis:

• Quadrant I: Top right
• Quadrant II: Top left
• Quadrant III: Bottom left
• Quadrant IV: Bottom right

Understanding which quadrant an angle's terminal side lies in helps in determining the signs of trigonometric functions related to that angle.

Negative Sine and Cosine

The concepts of negative sine and cosine values are crucial in determining the location of an angle's terminal side within the coordinate plane.

**Sine Function**
The sine function relates to the y-coordinate value of a point on the unit circle. Therefore, if \( \sin \theta < 0 \), the y-coordinate is negative. This condition is met in Quadrants III and IV.

**Cosine Function**
The cosine function is linked to the x-coordinate of a point on the unit circle. If \( \cos \theta < 0 \), it indicates that the x-coordinate is negative, which occurs in Quadrants II and III.

Combining these two, for both \( \sin \theta \) and \( \cos \theta \) to be negative simultaneously, we look where both x and y coordinates are negative. This occurs only in Quadrant III.

Coordination Geometry

Coordination geometry deals with understanding shapes and functions in a coordinate plane. It helps us grasp how trigonometric angles are positioned and understood. A co-ordinate plane divides into four quadrants, helping in visualizing angles.

The x-axis and y-axis intersect at the origin (0,0), creating four quadrants:

• Quadrant I: (+, +)
• Quadrant II: (-, +)
• Quadrant III: (-, -)
• Quadrant IV: (+, -)

The positive or negative signs of the trigonometric functions hinge on these quadrant arrangements. For angles in any quadrant, determining which trigonometric functions are positive or negative involves examining these coordinate signs.

Trigonometric Functions

Trigonometric functions are essential tools in geometry and trigonometry to relate angles with coordinate values. The basic trigonometric functions are sine, cosine, and tangent, each relating to the ratio of different sides of a right triangle.

**Sine ( sin)**: Represents the ratio of the length of the opposite side to the hypotenuse. In a unit circle, it indicates the y-coordinate of the point.

**Cosine (cos)**: Denotes the ratio of the adjacent side to the hypotenuse. On the unit circle, it shows the x-coordinate.

**Tangent (tan)**: This is the ratio of the sine and cosine values, or the opposite side over the adjacent side in a triangle.

Knowing which trigonometric functions are positive or negative in each quadrant helps define their properties more clearly. This greatly aids in solving various problems related to angles and their measurements in geometry.