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 relative to angle \( \theta \) at vertex R. The hypotenuse is the side opposite the right angle, which is \( PR = 53 \). The side opposite \( \theta \) is \( PQ = 28 \), and the adjacent side to \( \theta \) is \( QR \), which is unknown.
2. Use the Pythagorean Theorem to find the missing side \( QR \). The theorem states: \(QR^2 + PQ^2 = PR^2\) Substitute the known values: \(QR^2 + 28^2 = 53^2\)
3. Solve for \( QR^2 \): \(QR^2 = 53^2 - 28^2\) Then take the square root to find \( QR \): \(QR = \sqrt{53^2 - 28^2}\)
4. Once you have the length of \( QR \), find the six trigonometric functions of \( \theta \) using the definitions relative to angle \( \theta \): - \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{PQ}{PR} \) - \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{QR}{PR} \) - \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{PQ}{QR} \) - \( \csc \theta = \frac{1}{\sin \theta} = \frac{PR}{PQ} \) - \( \sec \theta = \frac{1}{\cos \theta} = \frac{PR}{QR} \) - \( \cot \theta = \frac{1}{\tan \theta} = \frac{QR}{PQ} \)
5. Substitute the known side lengths into these formulas to express each trigonometric function in terms of numbers. This completes the process of finding the missing side and the six trigonometric functions of \( \theta \).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. It is expressed as a² + b² = c², where c is the hypotenuse. This theorem helps find the missing side length when two sides are known.

Right Triangle Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—relate the angles of a right triangle to the ratios of its sides. For an angle θ, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. The reciprocal functions are cosecant, secant, and cotangent.

Identifying Sides Relative to an Angle

In a right triangle, the side opposite the right angle is the hypotenuse. For a given angle θ, the side directly opposite is the opposite side, and the side next to θ (but not the hypotenuse) is the adjacent side. Correctly identifying these sides is essential for applying trigonometric functions accurately.