Problem 52 - 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 \(\boldsymbol{\theta}^{\prime}\). $$\theta=225^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for the given angle of \(225^{\circ}\) is \(45^{\circ}\).

Step 1: Identify Quadrant

Firstly, determine the quadrant in which the terminal side of the given angle resides. In standard position, an angle of \(225^{\circ}\) falls in the third quadrant as it is greater than \(180^{\circ}\) but less than \(270^{\circ}\).

Step 2: Calculate the Reference Angle

Next, calculate the reference angle \(\theta^{\prime}\). Since the given angle \(\theta\) is in the third quadrant, the reference angle can be found by subtracting \(180^{\circ}\) from the given angle: \(\theta^{\prime} = \theta - 180^{\circ}\). Substituting our given value, \(\theta^{\prime} = 225^{\circ} - 180^{\circ} = 45^{\circ}\).

Step 3: Sketch the Angle

Finally, sketch the angle in standard position. The initial side is on the positive x-axis. Rotate counterclockwise by \(225^{\circ}\) to draw the terminal side in the third quadrant. Mark the reference angle \(\theta^{\prime}\) as \(45^{\circ}\). The angle \(\theta^{\prime}\) is the angle between the terminal side and the x-axis moving clockwise.

Step 4: Label the Reference Angle

In the sketch, label the reference angle \(\theta^{\prime}\) appropriately to indicate it clearly.

Key Concepts

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants. Each quadrant has different rules for the signs of trigonometric functions. This is because the axes divide the plane into sections:

• 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.

When you have an angle like \(225^{\circ}\), determining its quadrant is crucial. Since \(225^{\circ}\) is more than \(180^{\circ}\) but less than \(270^{\circ}\), it's in the third quadrant. Understanding these quadrants helps you to know the behavior of angles and their trigonometric values.

Special Angles in Trigonometry

Special angles, like \(45^{\circ}\), \(30^{\circ}\), and \(60^{\circ}\), often appear in trigonometry. They have known sine, cosine, and tangent values. These angles often arise from equilateral triangles and make sketching or solving simpler.

When finding a reference angle such as in \(225^{\circ}\), it simplifies calculations. In this case, since \(225^{\circ}\) is in the third quadrant, the reference angle \(\theta'\) is found by subtracting \(180^{\circ}\):
\[\theta' = 225^{\circ} - 180^{\circ} = 45^{\circ}\]

This shows that the trigonometric functions for \(225^{\circ}\) are related to those of \(45^{\circ}\), but with signs changed due to the quadrant.

Angle Sketching in Standard Position

Sketching an angle in standard position means you start from the positive x-axis, moving counterclockwise for positive angles. This is crucial when visualizing angles on the coordinate plane and understanding their properties.

For \(225^{\circ}\):

• Begin at the positive x-axis.
• Rotate counterclockwise to place the angle's terminal side into the third quadrant.

Visualizing the rotation helps in drawing accurate sketches. The reference angle \(\theta'\) is then drawn from this terminal side vertically to the x-axis, creating the \(45^{\circ}\) angle. Properly labeling these angles aids in understanding and using them correctly in problem-solving.