Determine whether each statement is true or false. If false, tell why. Use a calculator for Exercises 39 and 42.1 tan² 60° = sec² 60°

Verified step by step guidance

1. Recall the trigonometric identity: \( \tan^2 \theta + 1 = \sec^2 \theta \).
2. Substitute \( \theta = 60^\circ \) into the identity: \( \tan^2 60^\circ + 1 = \sec^2 60^\circ \).
3. Calculate \( \tan 60^\circ \) using a calculator or known values: \( \tan 60^\circ = \sqrt{3} \).
4. Calculate \( \tan^2 60^\circ \) by squaring the result from the previous step: \( (\sqrt{3})^2 = 3 \).
5. Calculate \( \sec 60^\circ \) using a calculator or known values: \( \sec 60^\circ = 2 \), then square it to find \( \sec^2 60^\circ = 4 \).

Key Concepts

Tangent and Secant Functions

The tangent function, denoted as tan, is the ratio of the opposite side to the adjacent side in a right triangle. The secant function, denoted as sec, is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Understanding these functions is crucial for evaluating the given statement involving tan² and sec².

Pythagorean Identity

The Pythagorean identity states that for any angle θ, tan²(θ) + 1 = sec²(θ). This relationship is fundamental in trigonometry and helps in verifying the truth of statements involving tangent and secant functions. Recognizing this identity allows for simplification and comparison of the two sides of the equation.

Calculator Use in Trigonometry

Using a calculator to find trigonometric values is essential for verifying statements involving angles. For example, calculating tan(60°) and sec(60°) will provide numerical values that can be squared and compared. Familiarity with calculator functions ensures accurate evaluations of trigonometric expressions.