Problem 15 - 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. $$ 90^{\circ} $$

Step-by-Step Solution

Answer

The coordinates are \((0, 1)\).

Step 1: Identify Angle Position

We start by analyzing the given angle of \(90^{\circ}\). This angle is in standard position, meaning its initial side is on the positive x-axis, and it rotates counterclockwise. At \(90^{\circ}\), the terminal side is perfectly aligned with the positive y-axis.

Step 2: Understand Unit Circle Properties

Recall that in a unit circle, which has a radius of 1, any point on its circumference has coordinates \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle in standard position. We need to find these coordinates for \(\theta = 90^{\circ}\).

Step 3: Calculate Coordinates

For \(\theta = 90^{\circ}\), use the definitions: \(\cos(90^{\circ}) = 0\) and \(\sin(90^{\circ}) = 1\). Thus, the coordinates of the point where the terminal side intersects the unit circle are \((0, 1)\).

Step 4: Rounding (if necessary)

The coordinates \((0, 1)\) are already precise and do not require rounding, as they represent exact values on the unit circle.

Key Concepts

Trigonometric Functions

When dealing with angles on the unit circle, trigonometric functions play a crucial role. They help in determining the precise location of points on the circle's circumference.

In the context of a unit circle:

• The sine function, denoted as \(\sin \theta\), gives the y-coordinate of a point on the circle.
• The cosine function, or \(\cos \theta\), provides the x-coordinate.
• The tangent function, \(\tan \theta\), is the ratio of \(\sin \theta\) over \(\cos \theta\), but this is more often used in different calculations.

The unit circle simplifies these trigonometric functions since every radius equals one. This means

• For any angle \(\theta\), \(\cos \theta\) calculates how far the point is from the origin horizontally.
• Meanwhile, \(\sin \theta\) measures the vertical distance from the origin.

When \(\theta = 90^{\circ}\), specifically

• \(\cos 90^{\circ} = 0\)
• \(\sin 90^{\circ} = 1\)

These functions tell us the coordinates of \((0, 1)\), confirming that at 90 degrees, the point lies directly on the y-axis.

Standard Position Angle

A standard position angle is a fundamental idea in trigonometry and geometry that helps us relate angles to their position on the unit circle.

Key features of standard position angles:

• They start with their vertex at the origin of a coordinate system—where the x and y axes meet.
• The initial side of the angle lies along the positive x-axis.
• The angle is measured by rotating the initial side counterclockwise towards the terminal side.

This kind of positioning makes it easier to consistently describe the coordinate of angles.
For example:

• A 90-degree angle in standard position rotates from the positive x-axis to align precisely with the positive y-axis.
• This alignment means the endpoint of the angle's terminal side is on the unit circle at coordinate \((0, 1)\).

Understanding this concept is crucial when studying angles as it provides a consistent reference that connects angle measurement with trigonometric values.

Right Angle

Right angles are a foundational concept in mathematics, representing a quarter of a full rotation, equating to 90 degrees.

Some characteristics of right angles include:

• Being the angle of intersection of two perpendicular lines.
• Forming an exact \(90^{\circ}\) angle, which is both visualized in geometric shapes such as squares and rectangles.
• Having significant importance in defining orthogonal vectors in physics and mathematics.

In terms of the unit circle, a right angle:

• Is located at the first quadrant to second quadrant boundary.
• It shows a perfect shift from the horizontal x-axis to a vertical position along the y-axis, achieving the coordinates \((0, 1)\).

Visualizing a right angle on the unit circle helps in understanding perpendicularity and its implications in more complex trigonometric concepts.