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. tan² 72°25' + 1 = sec² 72°25'

Verified step by step guidance

1. Convert the angle from degrees and minutes to decimal degrees: 72°25' = 72 + 25/60 degrees.
2. Use the identity \( \tan^2(\theta) + 1 = \sec^2(\theta) \) to verify the statement.
3. Calculate \( \tan(72.4167°) \) using a calculator and then square the result to find \( \tan^2(72.4167°) \).
4. Add 1 to \( \tan^2(72.4167°) \) to find \( \tan^2(72.4167°) + 1 \).
5. Calculate \( \sec(72.4167°) \) using a calculator and square the result to find \( \sec^2(72.4167°) \). Compare this with the previous result to determine if the statement is true or false.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One of the fundamental identities is the Pythagorean identity, which states that tan²(θ) + 1 = sec²(θ). Understanding these identities is crucial for verifying trigonometric statements and simplifying expressions.

Rounding Errors

Rounding errors occur when numerical values are approximated to a certain number of decimal places, leading to slight discrepancies in calculations. In trigonometry, using a calculator can introduce rounding errors, especially when dealing with angles in degrees and minutes. Recognizing the potential for these errors is important when evaluating the accuracy of trigonometric statements.

Degrees and Minutes

Angles can be measured in degrees, where one degree is 1/360 of a full circle, and in minutes, where one degree is divided into 60 minutes. The notation 72°25' indicates 72 degrees and 25 minutes. Converting between these units is essential for accurate trigonometric calculations, especially when using calculators that may require input in decimal degrees.