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: \(\tan^{2} 60^\circ + 1 = \sec^{2} 60^\circ\).
3. Calculate \(\tan 60^\circ\) and then square it: \(\tan 60^\circ = \sqrt{3}\), so \(\tan^{2} 60^\circ = (\sqrt{3})^{2} = 3\).
4. Calculate \(\sec 60^\circ\) and then square it: \(\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2\), so \(\sec^{2} 60^\circ = 2^{2} = 4\).
5. Compare \(\tan^{2} 60^\circ\) and \(\sec^{2} 60^\circ\): since \(3 \neq 4\), the statement \(\tan^{2} 60^\circ = \sec^{2} 60^\circ\) is false. The correct relation is \(\tan^{2} 60^\circ + 1 = \sec^{2} 60^\circ\).

Key Concepts

Trigonometric Functions and Their Values

Trigonometric functions like tangent (tan) and secant (sec) relate angles to ratios of sides in a right triangle or points on the unit circle. Knowing exact values for common angles such as 60° is essential; for example, tan 60° = √3 and sec 60° = 2. These values help evaluate and compare expressions accurately.

Pythagorean Identities

Pythagorean identities link squares of trigonometric functions, such as 1 + tan²θ = sec²θ. This identity is fundamental for verifying equations involving tan²θ and sec²θ, allowing one to determine if a given statement is true or false by substitution and simplification.

Use of Calculators for Verification

Calculators can compute approximate values of trigonometric functions to verify statements numerically. This is especially useful when exact values are complex or when confirming the truth of an equation involving trigonometric expressions, ensuring accuracy in determining true or false.