In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

Verified step by step guidance

1. Identify the sides of the right triangle: QR = 4 (adjacent), RP = 3 (opposite), and PQ (hypotenuse) is unknown.
2. Apply the Pythagorean Theorem: a^2 + b^2 = c^2, where a = 3, b = 4, and c is the hypotenuse.
3. Calculate the hypotenuse: 3^2 + 4^2 = c^2, which simplifies to 9 + 16 = c^2.
4. Solve for c: c = \sqrt{25}.
5. Find the six trigonometric functions of \( \theta \): \( \sin(\theta) = \frac{3}{5} \), \( \cos(\theta) = \frac{4}{5} \), \( \tan(\theta) = \frac{3}{4} \), \( \csc(\theta) = \frac{5}{3} \), \( \sec(\theta) = \frac{5}{4} \), \( \cot(\theta) = \frac{4}{3} \).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as a² + b² = c², where c is the hypotenuse. In this problem, it will be used to find the length of the missing side of triangle PQR.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions are sine (sin), cosine (cos), and tangent (tan), which are defined as sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. For triangle PQR, these functions will be calculated using the known side lengths and the angle θ.

Right Triangle Properties

Right triangles have specific properties that simplify calculations. The side opposite the right angle is the hypotenuse, while the other two sides are referred to as the legs. The relationships between the angles and sides in right triangles allow for the application of both the Pythagorean Theorem and trigonometric functions, making it easier to solve for unknown lengths and angles.