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. Step 1: Understand the problem. We need to compare \( \frac{1}{2} \sin 40^\circ \) with \( \sin \left( \frac{1}{2} \times 40^\circ \right) \).
2. Step 2: Calculate \( \frac{1}{2} \sin 40^\circ \). Use a calculator to find \( \sin 40^\circ \) and then multiply the result by \( \frac{1}{2} \).
3. Step 3: Calculate \( \sin \left( \frac{1}{2} \times 40^\circ \right) \). First, find \( \frac{1}{2} \times 40^\circ \) which is \( 20^\circ \), then use a calculator to find \( \sin 20^\circ \).
4. Step 4: Compare the two results from Step 2 and Step 3. Check if they are equal or if there is a difference.
5. Step 5: Consider rounding errors. If the results differ only in the last decimal place, it might be due to rounding errors. Otherwise, determine if the statement is true or false based on the comparison.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is defined for all real numbers and is periodic, with a range of values between -1 and 1. Understanding how the sine function behaves for different angles is crucial for evaluating expressions involving sine.

Angle Measurement

Angles can be measured in degrees or radians, and it is essential to be consistent with the unit used in calculations. In this context, 40° is an angle measurement in degrees, and converting it to radians may be necessary for certain calculations. Recognizing how to manipulate angles and their measurements is vital for solving trigonometric equations.

Rounding Errors

Rounding errors occur when numerical values are approximated to a certain number of decimal places, which can lead to discrepancies in calculations. In trigonometry, especially when using calculators, small differences in values can arise due to rounding, affecting the accuracy of results. Understanding the impact of rounding is important when comparing calculated values or determining the truth of mathematical statements.