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. ½ sin 40° = sin [½ (40°)]

Verified step by step guidance

1. Identify the two expressions given: the left side is \(\frac{1}{2} \sin 40^\circ\) and the right side is \(\sin \left( \frac{1}{2} \times 40^\circ \right)\).
2. Calculate the value of \(\sin 40^\circ\) using a calculator, then multiply that result by \(\frac{1}{2}\) to find the left side value.
3. Calculate the value of \(\frac{1}{2} \times 40^\circ = 20^\circ\), then find \(\sin 20^\circ\) using a calculator to get the right side value.
4. Compare the two results obtained from the left and right sides to see if they are approximately equal, considering possible minor differences due to rounding.
5. Conclude whether the statement \(\frac{1}{2} \sin 40^\circ = \sin \left( \frac{1}{2} \times 40^\circ \right)\) is true or false based on the comparison.

Key Concepts

Sine Function and Angle Measurement

The sine function relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. Angles are measured in degrees or radians, and the sine value depends on the angle's size, not on any linear scaling of the angle.

Properties of Trigonometric Functions

Trigonometric functions like sine are nonlinear, meaning that operations such as halving the angle inside the function do not equate to halving the function's value. For example, sin(40°)/2 is generally not equal to sin(20°).

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 when verifying equalities involving trigonometric expressions.