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 angles in the triangles: \( \pi/4 \) corresponds to 45° and \( \pi/6 \) corresponds to 30°.
2. For \( \tan(\pi/4) \), use the 45°-45°-90° triangle. The tangent of 45° is the ratio of the opposite side to the adjacent side, which is \( \frac{1}{1} \).
3. For \( \csc(\pi/6) \), use the 30°-60°-90° triangle. The cosecant of 30° is the reciprocal of the sine of 30°, which is \( \frac{2}{1} \).
4. Add the results from the previous steps: \( \tan(\pi/4) + \csc(\pi/6) = 1 + 2 \).
5. The expression simplifies to a single numerical value.

Key Concepts

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions include sine, cosine, and tangent, which are defined as ratios of the sides of a right triangle. For example, tangent (tan) is the ratio of the opposite side to the adjacent side. Understanding these functions is essential for evaluating expressions involving angles like π/4 and π/6.

Special Angles

Certain angles, such as 30°, 45°, and 60°, have known sine, cosine, and tangent values that are often used in trigonometry. For instance, tan(π/4) equals 1, and csc(π/6) equals 2. Familiarity with these special angles allows for quick evaluations of trigonometric expressions without needing a calculator.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate square roots from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable value that will result in a rational number in the denominator. This process is important in trigonometry to present answers in a standard form, making them easier to interpret and use in further calculations.