Use the given triangles to evaluate each expression. If...

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
$6 \tan \frac{\pi}{4}+\sin \frac{\pi}{3} \sec \frac{\pi}{6}$

Key Concepts

Special Triangles and Known Angle Values

Many trigonometric problems rely on the known values from special right triangles such as the 45°-45°-90° and the 30°-60°-90° triangles. These triangles provide exact values for trigonometric functions like sine, cosine, and tangent when the angle measure is among the commonly encountered ones (?/4, ?/3, ?/6, etc.). This principle is central in evaluating trigonometric expressions without the need for numerical approximation.

Trigonometric Functions and Reciprocal Functions

Understanding trigonometric functions such as sine and tangent, as well as their reciprocal functions like secant, is crucial. In problems involving expressions with multiple trigonometric functions, knowledge of the definitions (for example, secant being the reciprocal of cosine) enables the manipulation and evaluation of the expression. This concept also extends to understanding how these functions interact in combined expressions.

Rationalization of the Denominator

Rationalizing the denominator is an algebraic technique used to eliminate irrational numbers (often square roots) from the denominator of a fraction. This process involves multiplying the numerator and the denominator by an appropriate form of 1 to obtain an equivalent expression that is generally considered to be in a simplified or more standard form. It is an important concept for presenting results clearly in many areas of mathematics.

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