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 70° = 2 cos² 35° - 1

Verified step by step guidance

1. Recognize that the given equation resembles the double-angle identity for cosine, which states: \(\cos(2\theta) = 2\cos^{2}(\theta) - 1\).
2. Identify the angle \(\theta\) in the identity by comparing \(\cos 70^\circ\) with \(\cos(2\theta)\), so set \(2\theta = 70^\circ\) which gives \(\theta = 35^\circ\).
3. Rewrite the right side of the equation using the double-angle identity: \(2\cos^{2}(35^\circ) - 1\) should equal \(\cos(70^\circ)\) if the identity holds.
4. Use a calculator to find the numerical values of \(\cos 70^\circ\) and \(2\cos^{2} 35^\circ - 1\) separately, making sure your calculator is in degree mode.
5. Compare the two calculated values to determine if they are approximately equal, allowing for minor differences due to rounding errors, to conclude whether the statement is true or false.

Key Concepts

Cosine Double-Angle Identity

The cosine double-angle identity states that cos(2θ) = 2cos²(θ) - 1. This formula allows expressing the cosine of a double angle in terms of the cosine of the original angle, which is essential for verifying the given equation involving cos 70° and cos 35°.

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 trigonometric identities numerically.

Evaluating Trigonometric Expressions

Evaluating trigonometric expressions involves substituting angle values and calculating the result accurately. This skill is necessary to compare both sides of the equation and determine if the statement is true or false.