In Exercises 63–68, find the exact value of each expression. Do not use a calculator. tan(𝜋/3)/2 - 1/sec(𝜋/6)

Verified step by step guidance

1. Identify the given expression: \(\frac{\tan \frac{\pi}{3} - 1}{2 \sec \frac{\pi}{6}}\).
2. Recall the exact values of the trigonometric functions involved: \(\tan \frac{\pi}{3} = \sqrt{3}\), and \(\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}}\).
3. Find \(\cos \frac{\pi}{6}\) using the known exact value: \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), so \(\sec \frac{\pi}{6} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\).
4. Substitute these values back into the expression: \(\frac{\sqrt{3} - 1}{2 \times \frac{2}{\sqrt{3}}}\).
5. Simplify the denominator and then the entire fraction step-by-step, rationalizing denominators if necessary, to find the exact value.

Key Concepts

Exact Values of Special Angles

Certain angles like π/3 and π/6 have well-known exact trigonometric values derived from special triangles. For example, tan(π/3) equals √3, and sec(π/6) equals 2/√3. Knowing these values allows precise calculation without a calculator.

Definition and Relationship of Trigonometric Functions

Understanding how tangent and secant relate to sine and cosine is essential. Tangent is the ratio of sine to cosine (tan θ = sin θ / cos θ), while secant is the reciprocal of cosine (sec θ = 1 / cos θ). This helps in simplifying and evaluating expressions.

Simplifying Trigonometric Expressions

Combining and simplifying expressions involving trigonometric functions requires algebraic manipulation, such as finding common denominators or rationalizing denominators. This skill ensures the exact value is expressed in simplest form without approximations.