Problem 32 - Algebra 2

Question

Find each exact value. Use a sum or difference identity. $$ \sin 225^{\circ} $$

Step-by-Step Solution

Answer

The exact value of the sine of 225 degrees using the sum or difference identity is \(-\frac{\sqrt{2}}{2}\)

Step 1: Rewrite the angle.

Rewrite 225 degrees as a sum of the angles 180 degrees and 45 degrees, which are angles we know the sine values for. So, 225 degrees can be written as: \( 225° = 180° + 45° \)

Step 2: Apply the sine difference identity.

The sine difference identity is as follows: \( \sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B) \). Apply this identity to our equation with A as 180 degrees and B as 45 degrees. The formula now looks like: \( \sin (225°) = \sin(180°) \cos(45°) + \cos(180°) \sin(45°) \)

Step 3: Resolve the trigonometric values.

Generate the values of the sine and cosine. The sine of 180 degrees is 0, the cosine of 180 degrees is -1, the sine of 45 degrees is \( \frac{\sqrt{2}}{2} \), and the cosine of 45 degrees is \( \frac{\sqrt{2}}{2} \). Substitute these values into our equation. The equation now looks like: \( \sin (225°) = (0 * \frac{\sqrt{2}}{2}) - (\frac{\sqrt{2}}{2}) = -\frac{\sqrt{2}}{2} \)

Key Concepts

Sum and Difference Identities

In trigonometry, sum and difference identities play a crucial role in simplifying complex expressions and calculating trigonometric functions for various angles. These identities help us resolve trig equations involving sums or differences into simpler forms that can be easily evaluated.
These identities are particularly useful when working with angles that are not standard angles seen in reference tables, like 225° in our exercise. For sine, the sum and difference identity is expressed as:

• \( \sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B) \)
• \( \sin(A - B) = \sin(A) \cos(B) - \cos(A) \sin(B) \)

By using these identities, you can break down trigonometric calculations into manageable parts and efficiently find exact values. In the given example, the sine of 225° is calculated by decomposing it into simpler components (180° and 45°) using these identities, which simplifies the computation process remarkably.

Exact Trigonometric Values

Exact trigonometric values are specific values for trigonometric functions at specific angles that are known exactly because of their recurring occurrence in geometry, such as 0°, 30°, 45°, 60°, 90°, etc.
These values are typically found in basic trigonometric tables and are derived from geometric principles. Knowledge of exact values for angles like 180° and 45° is key to utilizing sum and difference identities effectively.
For example, in the exercise, we used:

• \( \sin(180°) = 0 \)
• \( \cos(180°) = -1 \)
• \( \sin(45°) = \frac{\sqrt{2}}{2} \)
• \( \cos(45°) = \frac{\sqrt{2}}{2} \)

Having these values memorized or easily accessible enables quick substitution into identities, significantly simplifying calculations. Understanding exact values aids in developing intuition for how trigonometric functions behave at key angles.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent express relationships between the angles and sides of triangles. They are foundational in geometry and trigonometry and have broad applications in various fields including engineering, physics, and computer graphics.
In the context of our problem, these functions are used to manipulate and calculate angles accurately. Here, the sine and cosine functions were crucial:

• Sine function \( \sin \theta \) measures the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine function \( \cos \theta \) measures the ratio of the adjacent side to the hypotenuse.

By understanding these functions, one can visualize the relationships between angles and sides, or even angles and coordinates in a unit circle. Mastery over trigonometric functions enables solving complex problems involving waves, oscillations, and rotations in a systematic manner.