In Exercises 29–36, find the length x to the nearest whole unit.

Verified step by step guidance

1. Identify the right triangle in the image and note the given angles and side length. The triangle has angles of 22° and 54°, and the adjacent side to the 22° angle is 200 units.
2. Use the trigonometric function tangent, which relates the opposite side to the adjacent side in a right triangle. The formula is $\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}$.
3. For the angle 22°, set up the equation using tangent: $\tan (22°)=\frac{x}{200}$, where $x$ is the length of the opposite side we need to find.
4. Solve for $x$ by multiplying both sides of the equation by 200: $x=200\times \tan (22°)$.
5. Calculate $\tan (22°)$ using a calculator and multiply by 200 to find the length $x$.

Key Concepts

Right Triangle Properties

A right triangle has one angle measuring 90 degrees, and the other two angles are complementary, meaning they add up to 90 degrees. In this problem, the angles are 22° and 54°, which confirms the triangle's right angle. Understanding these properties is essential for applying trigonometric functions correctly.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. For this triangle, the tangent function can be used to find the height (x) since it relates the opposite side to the adjacent side. Specifically, tan(22°) = x/200, which can be rearranged to solve for x.

Angle of Elevation

The angle of elevation is the angle formed by a horizontal line and the line of sight to an object above the horizontal. In this context, the angle of elevation from the base of the triangle to the top (height x) is 54°. This concept is crucial for determining the relationship between the height and the base when using trigonometric functions.