Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.


tan (15π/4)

Verified step by step guidance

1. Step 1: Understand the periodicity of the tangent function. The tangent function, \( \tan(\theta) \), has a period of \( \pi \). This means that \( \tan(\theta + n\pi) = \tan(\theta) \) for any integer \( n \).
2. Step 2: Simplify the angle \( \frac{15\pi}{4} \) by reducing it within the range of \( [0, \pi) \) using the periodicity.
3. Step 3: Calculate \( \frac{15\pi}{4} \mod \pi \) to find the equivalent angle within one period.
4. Step 4: Express \( \pi \) in terms of \( \frac{\pi}{4} \) to facilitate the calculation: \( \pi = \frac{4\pi}{4} \).
5. Step 5: Subtract \( \pi \) from \( \frac{15\pi}{4} \) as many times as needed to bring the angle within the range of \( [0, \pi) \), and then evaluate \( \tan \) at the resulting angle.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. They are periodic functions, meaning they repeat their values in regular intervals. Understanding these functions is essential for evaluating expressions involving angles, especially when the angles are expressed in radians.

Angle Reduction

Angle reduction involves simplifying angles to find equivalent angles within a standard range, typically between 0 and 2π radians. For example, to evaluate tan(15π/4), we can subtract multiples of 2π (or 8π/4) to find a coterminal angle, which makes it easier to compute the tangent value.

Undefined Values in Trigonometry

Certain values of trigonometric functions can be undefined, particularly when dealing with tangent and cotangent, which are undefined at odd multiples of π/2. Recognizing these points is crucial when evaluating expressions, as it helps determine whether a function can be computed or if it results in an undefined quantity.