Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The reference angle \( \theta' \) for the given special angle \( \theta = \frac{3 \pi}{4} \) is \( \frac{\pi}{4} \).

Step 1: Define The Angle in terms of \(\pi\)

We have the special angle \( \theta = \frac{3 \pi}{4} \) which lies in the second quadrant as it is larger than \( \pi/2 \) and smaller than \( \pi \). The standard position means the initial side of the angle is along the positive x-axis, and the terminal side is determined by the magnitude of \( \theta \) from the initial side in the counterclockwise direction.

Step 2: Calculate The Reference Angle

We have the special angle \( \theta = \frac{3 \pi}{4} \). To find the reference angle, we subtract from \( \pi \) because \( \theta \) lies in the second quadrant. Hence, the reference angle \( \theta' \) is \( \pi - \theta = \pi - \frac{3 \pi}{4} = \frac{\pi}{4} \). This is our acute reference angle which is formed by the terminal side of \( \theta \) and the x-axis.

Step 3: Drawing The Angle in Standard Position

The special angle \( \theta \) can be sketched in standard position, with its initial side (Base) on the positive x-axis and the terminal side (Hypotenuse) drawn counter-clockwise from the initial side by an angle of \( \frac{3 \pi}{4} \) (or 135°). Now, \( \frac{\pi}{4} \) (or 45°) segment drawn from the x-axis to the terminal side of \( \theta \) will give us the reference angle \( \theta' \).

Key Concepts

Reference Angle

A reference angle is an acute angle that provides a simplified way to work with angles. It is always measured between the terminal side of the angle and the x-axis. The reference angle is crucial because it helps understand the trigonometric functions' sign and value across different quadrants.
To find a reference angle, use these steps based on the quadrant:

• First Quadrant: The reference angle is the angle itself.
• Second Quadrant: Subtract the angle from \( \pi \).
• Third Quadrant: Subtract \( \pi \) from the angle.
• Fourth Quadrant: Subtract the angle from \( 2\pi \).

For example, for an angle \( \theta = \frac{3\pi}{4} \), which is in the second quadrant, the reference angle is \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \). This simplifies calculations and aids in understanding the angle's behavior.

Standard Position

In trigonometry, an angle is said to be in standard position when its initial side lies along the positive x-axis. The vertex of the angle is at the origin of the coordinate system. The most intuitive way to visualize an angle in standard position is to rotate from the initial side counterclockwise to form the terminal side.
If an angle is negative, the rotation is clockwise. For example, for \( \theta = \frac{3\pi}{4} \), the angle is positive, and its terminal side lies in the second quadrant as the angle extends from the positive x-axis counterclockwise. Sketching angles in standard position helps clearly see their relationship to the x-axis and identify which quadrant they lie in.

Special Angles

Special angles are specific angles commonly used in trigonometry because their sine, cosine, and tangent values are easy to memorize and frequently appear in problems. Common special angles include \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), and their multiples.
These angles have values like 30°, 45°, 60°, and 90° and their respective positions in radians. In our example, \( \theta = \frac{3\pi}{4} \) relates back to these values, as the reference angle \( \frac{\pi}{4} \) equals 45°. Recognizing special angles helps in evaluating trigonometric functions quickly.

Quadrants

The coordinate plane is divided into four quadrants, which help understand the sign of trigonometric functions. The quadrants are labeled counterclockwise starting from the positive x-axis:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, cosine is positive.

For an angle \( \theta = \frac{3\pi}{4} \), which lies in the second quadrant,

• the sign of sine is positive, and
• cosine is negative.

Understanding which quadrant an angle falls into determines the resulting sign of the trigonometric functions computed.