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 Pythagorean identity involving tangent and secant: \(\tan^2 \theta + 1 = \sec^2 \theta\).
2. Substitute \(\theta = 60^\circ\) into the identity to get \(\tan^2 60^\circ + 1 = \sec^2 60^\circ\).
3. Compare the given statement \(\tan^2 60^\circ = \sec^2 60^\circ\) with the identity. Notice that the identity includes an additional \(+1\) on the left side.
4. Calculate or recall the exact values: \(\tan 60^\circ = \sqrt{3}\), so \(\tan^2 60^\circ = 3\), and \(\sec 60^\circ = 2\), so \(\sec^2 60^\circ = 4\).
5. Since \(\tan^2 60^\circ = 3\) and \(\sec^2 60^\circ = 4\), the statement \(\tan^2 60^\circ = \sec^2 60^\circ\) is false because it ignores the \(+1\) in the Pythagorean identity.

Key Concepts

Pythagorean Identity involving Tangent and Secant

The identity tan²θ + 1 = sec²θ relates the tangent and secant functions. It means that the square of the tangent of an angle plus one equals the square of the secant of the same angle. This identity is fundamental for verifying trigonometric equations involving these functions.

Evaluating Trigonometric Functions at Specific Angles

To verify statements like tan² 60° = sec² 60°, you must calculate the exact or approximate values of tangent and secant at 60 degrees. Knowing that tan 60° = √3 and sec 60° = 2 helps in substituting and comparing both sides of the equation.

Use of Calculators for Trigonometric Verification

Calculators can provide decimal approximations of trigonometric values, which is useful for checking the truth of equations when exact values are complex or unknown. Using a calculator ensures accuracy in evaluating tan² 60° and sec² 60° for comparison.