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 given sides of the right triangle. Typically, you will have two sides given, and you need to find the third side. Label the sides as 'a', 'b', and 'c', where 'c' is the hypotenuse.
2. Use the Pythagorean Theorem: \( a^2 + b^2 = c^2 \). Substitute the known values into the equation to solve for the missing side.
3. Solve the equation for the missing side. If you are solving for the hypotenuse, rearrange to \( c = \sqrt{a^2 + b^2} \). If solving for a leg, rearrange to \( a = \sqrt{c^2 - b^2} \) or \( b = \sqrt{c^2 - a^2} \).
4. Once you have all three sides, use them to find the six trigonometric functions of \( \theta \): \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \), \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \), \( \csc(\theta) = \frac{1}{\sin(\theta)} \), \( \sec(\theta) = \frac{1}{\cos(\theta)} \), \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
5. Substitute the side lengths into the trigonometric function formulas to express each function in terms of the side lengths.

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 relationship can be expressed as a² + b² = c², where c represents the hypotenuse and a and b represent the other two sides. This theorem is fundamental for finding the length of a missing side in right triangles.

Trigonometric Functions

The six trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)—are ratios derived from the sides of a right triangle relative to one of its angles. For an angle θ, these functions are defined as follows: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. The reciprocal functions are csc(θ) = hypotenuse/opposite, sec(θ) = hypotenuse/adjacent, and cot(θ) = adjacent/opposite.

Right Triangle Properties

Right triangles have specific properties that distinguish them from other triangles, primarily that one angle measures 90 degrees. This unique angle allows the application of the Pythagorean Theorem and trigonometric functions. Additionally, the relationships between the angles and sides in right triangles are foundational for solving various problems in trigonometry, including those involving angle measures and side lengths.