Use P Q R with right angle R to find sinP, cosP, tanP sinQ,...

Use $\triangle P Q R$ with right angle $R$ to find $\sin P, \cos P, \tan P$
$\sin Q, \cos Q,$ and tan $Q .$ Express each ratio as a fraction, and as a decimal to the nearest hundredth.
$$p=\sqrt{6}, q=2 \sqrt{3}, \text { and } r=3 \sqrt{2}$$

Key Concepts

Right Triangle Geometry

A right triangle is a polygon with one 90-degree angle. The side opposite the right angle is called the hypotenuse, and the other two sides are known as the legs. Understanding the properties of right triangles is essential for solving problems in trigonometry, as the relationships between the side lengths lead directly to the definitions of the trigonometric ratios.

Trigonometric Ratios

Trigonometric ratios are functions that relate the angles of a triangle to the lengths of its sides. In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. These definitions form the foundation of much of trigonometry.

Complementary Angles in Right Triangles

In a right triangle, the two acute angles are complementary, meaning their sum is 90 degrees. This relationship implies that the sine of one angle is equal to the cosine of its complement and vice versa. Recognizing this connection can simplify many trigonometric problems and provides deeper insight into the behavior of trigonometric functions within right triangles.

Fraction and Decimal Conversions

In many trigonometric problems, it is useful to express ratios both in fractional form and in decimal form. Fractions provide exact values that are often preferable in theoretical work, while decimals are helpful for approximation and practical calculations. Converting between these forms is a key skill in ensuring that answers are both precise and understandable.