Problem 23 - Algebra 2 and Trigonometry

Question

In \(18-27,\) for each given angle, find a coterminal angle with a measure of \(\theta\) such that \(0 \leq \theta < 360\). $$ -270^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is \(90^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are those that share the same initial and terminal sides. To find a coterminal angle in the standard range \(0^{\circ} \leq \theta < 360^{\circ}\), we can add or subtract multiples of \360^{\circ}\ to the given angle.

Step 2: Finding the Coterminal Angle

The given angle is \theta = -270^{\circ}\. Since this angle is negative, we need a positive coterminal angle. We add \360^{\circ}\ to \theta\ until it falls within zero and \360^{\circ}\. Let’s add \360^{\circ}\: \(-270^{\circ} + 360^{\circ} = 90^{\circ}\).Now \theta = 90^{\circ}\ lies within the range \(0^{\circ} \leq \theta < 360^{\circ}\).

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in geometry. It involves determining the size of an angle in degrees. The degree is a unit of measurement used to measure angles, with a full circle being 360 degrees. This measurement is essential when discussing coterminal angles.
By understanding how angles are measured, students can explore how different angles relate to one another. Measuring angles involves comparing the turn made by a radius around a circle's center.
In this regard,

• A right angle measures 90 degrees.
• A straight angle measures 180 degrees.
• A full rotation is 360 degrees.

Understanding these basic angle measurements allows students to identify coterminal angles easily. Coterminal angles share the same position, meaning their measurements differ by a multiple of 360 degrees. This connection aids in solving problems where finding related angles is necessary.

Positive Angles

Positive angles are those that are measured in a counterclockwise direction from the initial side of the angle. In trigonometry, positive angles are significant because they help standardize angle measurements.
Angles measured counterclockwise from the x-axis are considered positive, while those measured clockwise are negative. For example, if you have a negative angle, like -270 degrees, you convert it to a positive angle by adding 360 degrees, which yields 90 degrees in this case.
This method of converting angles ensures that they fit into a standard interval, typically

• From 0 degrees to 360 degrees for degrees measurement.
• From 0 radians to 2π radians for radian measurement.

Understanding positive angles is crucial because it allows students to determine how angles relate to standard position and evaluate trigonometric functions.

Standard Position

The concept of an angle in standard position is crucial in trigonometry and geometry. An angle is said to be in standard position when its vertex is located at the origin of a coordinate plane, and its initial side lies on the positive x-axis. This position becomes a reference point to determine angles and their trigonometric values.
Any rotation from this position can be measured as an angle. Angles in standard position help standardize how we analyze angles and perform a range of geometric and trigonometric computations. This is particularly useful when calculating coterminal angles.

• By using the concept of standard position, we can easily determine the coterminal angles that fall within one full rotation of the circle.
• For example, an angle starting from the positive x-axis and moving counterclockwise by 90 degrees remains in that standardized range, making analysis straightforward.

Through this understanding, students can compute precise angle measures and transitions without confusion.