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. First, recognize the Pythagorean identity involving tangent and secant: \(\tan^2 \theta + 1 = \sec^2 \theta\). This identity holds true for any angle \(\theta\) where these functions are defined.
2. Convert the angle from degrees and minutes to decimal degrees for calculator input. Since 1 minute is \(\frac{1}{60}\) degrees, calculate \(72°25' = 72 + \frac{25}{60}\) degrees.
3. Calculate \(\tan(72.4167°)\) using the calculator, then square the result to find \(\tan^2(72.4167°)\).
4. Add 1 to the value obtained in the previous step to compute \(\tan^2(72.4167°) + 1\).
5. Calculate \(\sec(72.4167°)\) by finding \(\frac{1}{\cos(72.4167°)}\) using the calculator, then square this value to get \(\sec^2(72.4167°)\). Finally, compare this result with the value from step 4 to determine if the statement is true or false, considering minor differences due to rounding.

Key Concepts

Pythagorean Identity for Tangent and Secant

The identity tan²θ + 1 = sec²θ is a fundamental trigonometric identity derived from the Pythagorean theorem. It relates the square of the tangent of an angle to the square of the secant of the same angle, holding true for all angles where these functions are defined.

Angle Conversion and Notation

Angles given in degrees and minutes (e.g., 72°25') must be converted to decimal degrees or radians for calculator use. One minute equals 1/60 of a degree, so 72°25' equals 72 + 25/60 degrees, ensuring accurate input for trigonometric calculations.

Rounding Errors in Calculator Computations

Calculators approximate trigonometric values, which can cause minor differences in the last decimal places. Understanding that slight discrepancies do not invalidate identities is important when verifying trigonometric statements numerically.