Problem 22 - Algebra 2 and Trigonometry

Question

In \(15-22\) , for each given angle in standard position, determine to the nearest tenth the coordinates of the point where the terminal side intersects the unit circle. $$ -45^{\circ} $$

Step-by-Step Solution

Answer

The coordinates are approximately (0.7, -0.7).

Step 1: Determine Reference Angle

The reference angle for a given angle is the positive acute angle that the terminal side makes with the x-axis. To find the reference angle for \(-45^{\circ}\), note that it's already an acute angle. So, the reference angle is \(45^{\circ}\).

Step 2: Use Unit Circle Definition

On the unit circle, every angle \(\theta\)'s point of intersection \( (x, y) \) corresponds to the coordinates \((\cos \theta, \sin \theta)\).

Step 3: Calculate Cosine and Sine

For \(-45^{\circ}\), calculate the cosine and sine. \(\cos(-45^{\circ}) = \cos(45^{\circ}) = \frac{\sqrt{2}}{2}\) and \(\sin(-45^{\circ}) = -\sin(45^{\circ}) = -\frac{\sqrt{2}}{2}\).

Step 4: Point Coordinates on Unit Circle

The coordinates where the terminal side of \(-45^{\circ}\) intersects the unit circle are \(\left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right)\).

Key Concepts

Standard Position Explained

When working with angles in trigonometry, particularly on the unit circle, it is essential to understand what is meant by "standard position." An angle is said to be in standard position when its vertex is at the origin of a coordinate system and its initial side lies along the positive x-axis.
The direction of rotation determines the sign of the angle: a counterclockwise rotation from the initial side is positive, while a clockwise rotation is negative.
For example, in the case of \(-45^{\circ}\), the angle is generated by rotating 45 degrees in the clockwise direction from the positive x-axis.

Understanding the Reference Angle

A reference angle allows you to simplify trigonometric calculations by expressing angles in terms of their acute counterparts. It is defined as the smallest angle formed by the terminal side of the given angle and the x-axis.
Reference angles are always positive and acute, meaning they are always less than 90 degrees.
In the scenario where the angle is \(-45^{\circ}\), since it is already an acute angle, the reference angle is \(45^{\circ}\) itself.

• The reference angle helps in calculating trigonometric functions easily.
• It applies to both positive and negative angles, ensuring consistency.

Cosine and Sine on the Unit Circle

On the unit circle, any angle \( \theta \) is linked to a unique point, and the coordinates of this point are given as \((\cos \theta, \sin \theta)\).
These coordinates directly relate to the cosine and sine of the angle.

• For \( \cos(-45^{\circ}) \), it equals \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \).
• For \( \sin(-45^{\circ}) \), using the property of sine being an odd function, it equals \(-\sin(45^{\circ}) = -\frac{\sqrt{2}}{2} \).

Thus, the coordinates of the point where the terminal side of \(-45^{\circ}\) intersects the unit circle are \( \left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \).

Coordinate System and its Importance

The coordinate system is crucial for understanding angles and their trigonometric functions in the context of the unit circle. It provides a framework where angles in standard position can be clearly visualized and measured.
This system consists of two perpendicular lines (axes) meeting at an origin point, typically labeled as (0, 0).

• The horizontal axis is the x-axis, and the vertical axis is the y-axis.
• Angles in standard position originate from the positive x-axis and move along the unit circle.

By positioning angles within this coordinate system, we can precisely pinpoint their intersection on the circle and the corresponding sine and cosine values.