Use a calculator to evaluate each expression. 2 sin 25°13' cos 25°13' - sin 50°26'

Verified step by step guidance

1. First, recognize that the angles are given in degrees and minutes. Convert the angle 25°13' into decimal degrees if needed, or use the angle directly in your calculator by inputting degrees and minutes appropriately.
2. Recall the double-angle identity for sine: \(\sin(2\theta) = 2 \sin \theta \cos \theta\). Notice that the expression \(2 \sin 25°13' \cos 25°13'\) matches the left side of this identity with \(\theta = 25°13'\).
3. Rewrite the expression \(2 \sin 25°13' \cos 25°13'\) as \(\sin(2 \times 25°13')\), which simplifies to \(\sin 50°26'\).
4. Substitute this back into the original expression to get \(\sin 50°26' - \sin 50°26'\).
5. Since the two terms are the same, their difference is zero. This shows the expression simplifies to zero without needing further calculation.

Key Concepts

Trigonometric Functions and Their Values

Sine and cosine are fundamental trigonometric functions that relate angles to ratios of sides in a right triangle. Understanding how to evaluate these functions for given angles, including those expressed in degrees and minutes, is essential for solving the expression.

Angle Conversion and Notation

Angles given in degrees and minutes (e.g., 25°13') must be accurately interpreted or converted to decimal degrees for calculator input. One minute equals 1/60 of a degree, so converting ensures precise evaluation of trigonometric functions.

Trigonometric Identities

The expression involves terms like 2 sin A cos A and sin 2A, which are connected by the double-angle identity: sin 2A = 2 sin A cos A. Recognizing and applying this identity simplifies the expression and aids in verifying the result.