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 triangle: hypotenuse PR = 10, one leg QR = 6, and the missing side is PQ.
2. Use the Pythagorean Theorem: a^2 + b^2 = c^2, where c is the hypotenuse.
3. Substitute the known values into the equation: 6^2 + PQ^2 = 10^2.
4. Solve for PQ: PQ^2 = 10^2 - 6^2.
5. Calculate the six trigonometric functions of θ using the sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent, and their reciprocals.

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. This theorem is essential for finding the length of the missing side in right triangles.

Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—relate the angles of a triangle to the ratios of its sides. For angle θ in a right triangle, sine(θ) = opposite/hypotenuse, cosine(θ) = adjacent/hypotenuse, and tangent(θ) = opposite/adjacent. Understanding these functions is crucial for solving problems involving angles and side lengths.

Right Triangle Properties

Right triangles have specific properties that simplify calculations, including the relationship between the angles and sides. The sum of the angles in any triangle is 180 degrees, and in a right triangle, one angle is always 90 degrees. This property allows for the use of trigonometric ratios and the Pythagorean Theorem to find unknown side lengths and angles.