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. cos(30° + 20°) = cos 30° + cos 20°

Verified step by step guidance

1. Recall the cosine addition formula: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).
2. Apply the formula to the left side of the equation: \(\cos(30^\circ + 20^\circ) = \cos 30^\circ \cos 20^\circ - \sin 30^\circ \sin 20^\circ\).
3. Calculate the right side of the equation as given: \(\cos 30^\circ + \cos 20^\circ\).
4. Use a calculator to find the numerical values of both sides: compute \(\cos 30^\circ \cos 20^\circ - \sin 30^\circ \sin 20^\circ\) and \(\cos 30^\circ + \cos 20^\circ\) separately.
5. Compare the two results to determine if the original statement is true or false, keeping in mind that small differences in the last decimal place may be due to rounding errors.

Key Concepts

Cosine of a Sum Formula

The cosine of the sum of two angles is given by cos(A + B) = cos A cos B - sin A sin B. This identity is fundamental for evaluating expressions involving the cosine of angle sums and helps verify if an equation involving cos(30° + 20°) is true.

Properties of Trigonometric Functions

Trigonometric functions like cosine are nonlinear and do not distribute over addition, meaning cos(A + B) is not equal to cos A + cos B. Understanding this property is essential to avoid common misconceptions when comparing trigonometric expressions.

Rounding Errors in Calculator Computations

Calculators approximate trigonometric values, which can cause minor differences in the last decimal places. Recognizing that small discrepancies may arise from rounding helps in correctly interpreting the truth value of trigonometric statements.