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 expression \(\sin 10^\circ + \sin 10^\circ\). Since both angles are the same, \(A = B = 10^\circ\), so the expression becomes \(2 \sin \left( \frac{10^\circ + 10^\circ}{2} \right) \cos \left( \frac{10^\circ - 10^\circ}{2} \right)\).
3. Simplify the angles inside the sine and cosine: \(2 \sin 10^\circ \cos 0^\circ\).
4. Recall that \(\cos 0^\circ = 1\), so the expression simplifies to \(2 \sin 10^\circ\).
5. Compare this result to \(\sin 20^\circ\) by calculating both \(2 \sin 10^\circ\) and \(\sin 20^\circ\) using a calculator to determine if the original statement \(\sin 10^\circ + \sin 10^\circ = \sin 20^\circ\) is true or false.

Key Concepts

Sine Function and Angle Addition

The sine function relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. The sine of a sum of two angles is not simply the sum of their sines; instead, sin(a + b) = sin a cos b + cos a sin b. This identity is crucial to evaluate whether sin 10° + sin 10° equals sin 20°.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angle values. Understanding identities like the sine addition formula helps verify or disprove statements involving sums of sine values. These identities allow simplification and comparison of expressions.

Calculator Use and Rounding Errors

Using a calculator to evaluate trigonometric expressions involves approximations that can cause small rounding errors, especially in decimal places. Recognizing that minor differences may arise due to rounding helps interpret results accurately when verifying trigonometric statements.