CONCEPT PREVIEW Match each equation in Column I with the appropriate right triangle in Column II. In each case, the goal is to find the value of x. I II1. A. B. C. 2. 3. x = 5 tan 38° D. E. F. 4.5.6.

Verified step by step guidance

1. Identify the equation given in Column I: \( x = 5 \tan 38^\circ \).
2. Recall the definition of the tangent function in a right triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
3. In the equation \( x = 5 \tan 38^\circ \), recognize that 5 represents the length of the adjacent side to the angle 38°.
4. Set up the equation using the tangent definition: \( \tan 38^\circ = \frac{x}{5} \).
5. Solve for \( x \) by multiplying both sides by 5: \( x = 5 \times \tan 38^\circ \).

Key Concepts

Trigonometric Ratios

Trigonometric ratios are relationships between the angles and sides of a right triangle. The primary ratios are sine, cosine, and tangent, defined as the ratios of the lengths of the sides opposite, adjacent, and hypotenuse to a given angle. Understanding these ratios is essential for solving problems involving right triangles, as they allow for the calculation of unknown side lengths or angles.

Tangent Function

The tangent function is one of the primary trigonometric functions, defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. It is often represented as tan(θ) = opposite/adjacent. In the context of the given question, using the tangent function allows for the calculation of the side lengths when an angle and one side length are known.

Right Triangle Properties

Right triangles have specific properties that simplify the analysis of their angles and sides. The Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, is fundamental. Additionally, the angles in a right triangle always sum to 180 degrees, with one angle being 90 degrees, which is crucial for determining unknown angles or sides in trigonometric problems.