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 II 1. A. B. C. 2. 3. D. E. F. 4. 5. x = 5 sin 38° 6.

Verified step by step guidance

1. Identify the trigonometric function used in the equation. In this case, it is the sine function: $\sin \theta$.
2. Recall the definition of the sine function in a right triangle: $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$.
3. In the equation $x=5\sin {38}^{\circ }$, the number 5 represents the hypotenuse, and $\sin {38}^{\circ }$ represents the ratio of the opposite side to the hypotenuse.
4. To find the value of $x$, which is the length of the opposite side, multiply the hypotenuse (5) by $\sin {38}^{\circ }$.
5. Use a calculator to find $\sin {38}^{\circ }$ and then multiply it by 5 to solve for $x$.

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 the angles. For example, in a right triangle, the sine of an angle is the length of the opposite side divided by the hypotenuse. Understanding these ratios is essential for solving problems involving right triangles.

Sine Function

The sine function is one of the fundamental trigonometric functions, defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is often abbreviated as 'sin' and is crucial for calculating unknown side lengths or angles. For instance, in the equation x = 5 sin 38°, the sine of 38° is used to find the length of the side opposite this angle when the hypotenuse is known to be 5.

Right Triangle Properties

Right triangles have specific properties that make them unique, including the Pythagorean theorem, which states that the square of the hypotenuse equals the sum of the squares of the other two sides. Additionally, the angles in a right triangle always sum to 180 degrees, with one angle being exactly 90 degrees. These properties are essential for solving various trigonometric problems, including those that require finding unknown side lengths or angles.