Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-6.5$$

Step-by-Step Solution

Answer

The reference angle \( \theta^{\prime} \) corresponding to the given angle \( \theta = -6.5 \) is 6.5.

Step 1: Identify the given angle

The angle given in the problem is \(\theta = -6.5\). The negative sign indicates that the angle is measured in the clockwise direction.

Step 2: Calculate the reference angle

The reference angle \( \theta^{\prime} \) for an angle \( \theta = -6.5 \) is the absolute value of \( \theta \). So, we calculate \( \theta^{\prime} \) as follows: \( \theta^{\prime} \) = | -6.5 | = 6.5.

Step 3: Sketch the angle in standard position

To visually describe this: \n\n Imagine a right-handed x-y coordinate system. The positive x-axis represents the initial side of the angle. The angle of -6.5 is measured in the clockwise direction (because it's negative) from the positive x-axis. \n\n Reference angle \(\theta^{\prime}\) of 6.5 is simply the angle measured counterclockwise from the negative x-axis to the terminal side of \(\theta\). So, it coincides with \(\theta\) itself in this case.

Step 4: Label the reference angle

Label the obtained reference angle \( \theta^{\prime} \) on the sketch.

Key Concepts

standard position

Angles in standard position are a fundamental concept in trigonometry. When an angle is in standard position, the vertex of the angle is at the origin of the coordinate plane and its initial side lies along the positive x-axis.

The other side of the angle, known as the terminal side, will vary depending on the angle's measurement.

When sketching angles, always start from the positive x-axis for standard position and move towards the terminal side:

• Positive angles are measured counterclockwise from the positive x-axis.
• Negative angles are measured clockwise.

In the context of this exercise, it means that the angle \( \theta = -6.5 \) is first drawn beginning from the positive x-axis and stretching in the clockwise direction to measure 6.5 degrees.

negative angle

Understanding negative angles requires grasping how they are measured differently from positive angles. Unlike positive angles that are measured counterclockwise from the positive x-axis, negative angles sweep in the clockwise direction.

This can sometimes be confusing because we often visualize angles as moving counterclockwise. Yet, both directions serve a purpose when defining angles in trigonometry.

For example:

• A \( -30^{\circ} \) angle moves from the x-axis in a clockwise rotation, arriving at its terminal position.
• The direction of the rotation informs us of the sign, with clockwise being negative and counterclockwise being positive.

In practical terms, negative angles allow us to measure angles that go the opposite path, which can be helpful in calculations and applications that depend on direction, like in physics and engineering.

absolute value

The concept of absolute value is crucial in determining reference angles. Absolute value signifies the non-negative magnitude of a number, disregarding its sign.
In the context of angles, it helps us convert any angle to its equivalent positive measure, and this positive angle is called the reference angle.

For instance:

• If \( \theta = -6.5 \), the reference angle \( \theta^{\prime} \) is the absolute value — that is \( | -6.5 | = 6.5 \).
• The absolute value helps in focussing on the size or measure of the angle rather than its direction.

In this problem, the use of absolute value allowed us to find the positive representation of the angle without considering its initial negative direction.

This magnitude provides a consistent measure that is useful in both solving math problems and understanding physical rotations.