Problem 19 - Algebra 2

Question

The measure \(\boldsymbol{\theta}\) of an angle in standard position is given. Find the exact values of \(\cos \theta\) and \(\sin \theta\) for each angle measure. \(-\frac{\pi}{2}\) radians

Step-by-Step Solution

Answer

\(\cos\theta = 0\), \(\sin\theta = -1\)

Step 1: Identity of Angle Measure

The given angle is -\frac{\pi}{2} radians. This angle is located in the 4th quadrant of the unit circle, oriented clockwise from the positive x-axis.

Step 2: Identify the x/y coordinates in the Unit Circle

The point of intersection of the terminal side with the unit circle has coordinates (0,-1) in the 4th quadrant.

Step 3: Calculate Cosine and Sine Value

The \(\cos\theta\) (or x-coordinate) is 0 and the \(\sin\theta\) (or y-coordinate) is -1.

Key Concepts

unit circle

The unit circle is a fundamental concept in trigonometry and geometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. This circle is crucial because it simplifies the understanding of trigonometric functions and their values. The coordinates of any point on the unit circle,

• (x, y), represent the cosine and sine of the angle formed by a line from the origin to the point.
• The x-coordinate corresponds to the cosine (\( ext{cos} heta\)), while the y-coordinate corresponds to the sine (\( ext{sin} heta\)).

The angles in the unit circle are usually measured starting from the positive x-axis and can be positive or negative, indicating counterclockwise and clockwise directions respectively. Understanding these coordinates and how they relate to angles is essential for grasping other trigonometric concepts.

angle measurement

Angles are measured in degrees and radians,

• Degrees are more common in everyday contexts, while radians are often used in higher mathematics.
• One full revolution around a circle equals 360 degrees or \(2\pi\) radians.
• To convert degrees to radians, use the formula: radians = degrees (\(\times \frac{\pi}{180}\)).

The angle \(-\frac{\pi}{2}\) radians represents a half-turn starting clockwise from the positive x-axis, positioning it directly on the negative y-axis. Understanding angle measurements in both systems enhances your ability to navigate between different mathematical problems and contexts.

sine and cosine

Sine and cosine are foundational trigonometric functions. They describe relationships within right triangles and circular movements. On the unit circle,

• sine ( \(\sin\theta\) ) is the y-coordinate, representing the vertical distance of a point from the x-axis.
• cosine ( \(\cos\theta\) ) is the x-coordinate, representing the horizontal distance of a point from the y-axis.

For the angle \(-\frac{\pi}{2}\) radians, the coordinates on the unit circle are \((0, -1)\). Therefore,

• sine is equal to \(-1\) because it corresponds to the y-coordinate.
• cosine is equal to \(0\) since it corresponds to the x-coordinate.

These values indicate the position of a point on the unit circle concerning an angle measured from the positive x-axis.

radian measure

Radian measure is a way to quantify angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts,

• radians relate the angle size to the length of the arc it subtends.
• One radian is the angle created when the arc length equals the radius of the circle.

This measure allows for more straightforward calculations and expressions in calculus and mathematical analysis. A special case for radian measure is \(-\frac{\pi}{2}\). In this instance, the angle is

• expressed negatively, indicating a clockwise direction, thus highlighting how radians can fluidly express dynamic and static movements on the unit circle.

Embracing radians helps in understanding complex mathematical models and their real-world applications.