Problem 12 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function. $$\sin 225^{\circ}$$

Step-by-Step Solution

Answer

\( \sin 225^{\circ} = -\frac{\sqrt{2}}{2} \).

Step 1: Determine the Reference Angle

To find \( \sin 225^{\circ} \), we first determine the reference angle. Since 225° is in the third quadrant, subtract 180° from 225° to find the reference angle: \[ 225^{\circ} - 180^{\circ} = 45^{\circ} \] Thus, the reference angle is 45°.

Step 2: Identify the Sine Value of the Reference Angle

The sine of the reference angle 45° is a well-known trigonometric value: \[ \sin 45^{\circ} = \frac{\sqrt{2}}{2} \]

Step 3: Determine the Sign of Sine in the Third Quadrant

In the third quadrant, the sine function is negative. Hence, for an angle in the third quadrant, like 225°, the sine will be negative.

Step 4: Calculate the Sine of 225°

Combining the results from Step 2 and Step 3, we find that: \[ \sin 225^{\circ} = -\frac{\sqrt{2}}{2} \] Therefore, the exact value of \( \sin 225^{\circ} \) is negative because it is in the third quadrant.

Key Concepts

Understanding Reference Angles

In trigonometry, reference angles are essential for simplifying the process of evaluating trigonometric functions. A reference angle is the acute angle that a given angle makes with the horizontal axis. It is always measured between 0° and 90°.
To find the reference angle for angles above 180° and up to 360°, like 225°, subtract 180° from the given angle. This process helps to find a corresponding acute angle, which can make calculations more straightforward.

• For angle 225°, the calculation for the reference angle is: \( 225^{\circ} - 180^{\circ} = 45^{\circ} \).
• Thus, the reference angle for \( 225^{\circ} \) is \( 45^{\circ} \).

By understanding the concept of reference angles, we can easily evaluate trigonometric functions using known values instead of recalculating from scratch each time.

The Sine Function Simplified

The sine function is one of the fundamental trigonometric functions and is used to determine the y-coordinate of a point on the unit circle. For angles up to 90°, the sine value is straightforward since it reflects the y-coordinate directly.
Mathematically, for an angle \( \theta \), the sine function is defined as:
\[ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \]
However, the common angles within 90° have well-known sine values. For instance:

• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• This value is pivotal in many trigonometric calculations, including those involving angle transformations using reference angles.

Using Sine for Larger Angles

When working with angles larger than 90°, the sine value can be determined based on the reference angle. This is particularly useful when the angle falls in a quadrant where the sine values are negative, such as in the third quadrant.

Trigonometric Functions in the Third Quadrant

The unit circle's structure makes it important to analyze how trigonometric functions behave in each quadrant. The third quadrant, where angles range from 180° to 270°, poses a unique aspect of trigonometric functions.
In this quadrant, both the x and y coordinates of any point are negative, which affects the signs of sine, cosine, and tangent functions. For the sine function:

• The sine of any angle in the third quadrant is negative because it reflects the negative y-coordinate of the unit circle.
• Therefore, even though the reference angle for \( 225^{\circ} \) is \( 45^{\circ} \), because \( 225^{\circ} \) is in the third quadrant, its sine value is negative: \( -\frac{\sqrt{2}}{2} \).

Understanding this property of the third quadrant is crucial when determining the sign of trigonometric function values for angles greater than 180°.