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.
$$p=\frac{3}{2}, q=\frac{3 \sqrt{3}}{2}, \text { and } r=3$$

Key Concepts

Fraction to Decimal Conversion and Rounding

This concept involves the process of converting a ratio expressed as a fraction into its decimal form and rounding it to a specified degree of accuracy, such as to the nearest hundredth. This is important for presenting answers in both exact and approximate forms, ensuring precision and clarity in mathematical communication.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse’s length is equal to the sum of the squares of the other two sides. This theorem is crucial for finding unknown side lengths when only two sides are known, and it underpins many calculations involving trigonometric ratios.

Right Triangle Trigonometry

This concept focuses on triangles that include a 90-degree angle and explores the relationships between the lengths of their sides and their angles. Understanding the basic structure of right triangles is essential for applying trigonometric ratios and solving various geometric problems.

Trigonometric Ratios (Sine, Cosine, and Tangent Definitions)

Sine, cosine, and tangent are fundamental ratios in right triangle trigonometry. They are defined with respect to an acute angle in the triangle: sine is the ratio of the length of the side opposite the angle to the length of the hypotenuse, cosine is the ratio of the length of the adjacent side to the hypotenuse, and tangent is the ratio of the length of the opposite side to the adjacent side.

Complementary Angle Relationships

In any right triangle, the two acute angles are complementary, meaning they add up to 90 degrees. This relationship implies that the sine of one acute angle is equal to the cosine of the other, and vice versa, which is useful when converting or comparing trigonometric ratios between different angles in the triangle.

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