In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

tan 𝜋/4 + csc 𝜋/6

Verified step by step guidance

1. Identify the trigonometric functions to evaluate: \(\tan \frac{\pi}{4}\) and \(\csc \frac{\pi}{6}\).
2. Recall that \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\) and \(\csc \theta = \frac{1}{\sin \theta}\).
3. From the first triangle (45°-45°-90°), find \(\tan \frac{\pi}{4}\) by using the sides opposite and adjacent to the 45° angle: \(\tan \frac{\pi}{4} = \frac{1}{1}\).
4. From the second triangle (30°-60°-90°), find \(\sin \frac{\pi}{6}\), which is the ratio of the side opposite 30° to the hypotenuse: \(\sin \frac{\pi}{6} = \frac{1}{2}\), then calculate \(\csc \frac{\pi}{6} = \frac{1}{\sin \frac{\pi}{6}}\).
5. Add the two values: \(\tan \frac{\pi}{4} + \csc \frac{\pi}{6}\), and if necessary, rationalize the denominator to express the value without a square root in the denominator.

Key Concepts

Special Right Triangles

Special right triangles, such as the 45°-45°-90° and 30°-60°-90° triangles, have fixed side length ratios. For a 45°-45°-90° triangle, the sides are in the ratio 1:1:√2. For a 30°-60°-90° triangle, the sides are in the ratio 1:√3:2. These ratios help quickly determine trigonometric values without using a calculator.

Trigonometric Functions (tan and csc)

The tangent function (tan) of an angle in a right triangle is the ratio of the opposite side to the adjacent side. The cosecant function (csc) is the reciprocal of sine, defined as the hypotenuse divided by the opposite side. Understanding these definitions allows evaluation of expressions like tan(π/4) and csc(π/6) using triangle side lengths.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This process simplifies expressions and is often required for final answers in trigonometry problems to present values in a standard form.