Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. sin 10° + sin 10° = sin 20°

Verified step by step guidance

1. Recall the trigonometric identity for the sum of sines: \(\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).
2. Apply this identity to the left side of the equation where both angles are 10°, so \(A = 10^\circ\) and \(B = 10^\circ\).
3. Calculate the left side using the identity: \(\sin 10^\circ + \sin 10^\circ = 2 \sin \left( \frac{10^\circ + 10^\circ}{2} \right) \cos \left( \frac{10^\circ - 10^\circ}{2} \right)\).
4. Simplify the expression inside the sine and cosine functions: \(2 \sin 10^\circ \cos 0^\circ\).
5. Since \(\cos 0^\circ = 1\), the left side simplifies to \(2 \sin 10^\circ\). Compare this value with \(\sin 20^\circ\) by calculating both using a calculator to determine if the original statement is true or false.

Key Concepts

Basic Sine Function Properties

The sine function relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. It is important to understand that sine values do not generally add linearly, meaning sin(a) + sin(b) is not equal to sin(a + b).

Sum-to-Product Identities

Sum-to-product formulas transform sums of sines or cosines into products, such as sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2). These identities help simplify or verify trigonometric expressions and are essential for evaluating sums like sin 10° + sin 10°.

Use of Calculators and Rounding Errors

Calculators approximate trigonometric values, which can cause minor differences in the last decimal places due to rounding. Understanding this helps interpret results correctly, especially when verifying if an equation like sin 10° + sin 10° = sin 20° is true or false.