Problem 91 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. A 45 -foot-long hill makes an angle of \(20^{\circ}\) with the level ground at the bottom of it. After a big snowfall, people like to sled down the hill. How high above the surrounding level ground do they begin their descent?

Step-by-Step Solution

Answer

The height of the hill above the surrounding level ground is the result of \( 45 * \sin20^\circ \), approximately 15.3783 feet (only needed if rounding to four decimal places), from the level ground to the top of the hill.

Step 1: Identify the given values

In the described scenario, a hill forms the hypotenuse of a right-angled triangle. The length of the hill, which is the hypotenuse of the triangle, is 45 feet and the angle at the foot of the hill, \( \theta \), is \( 20^\circ \). The task is to find the opposite side of the right triangle, which is the height of the hill above level ground.

Step 2: Use sine formula

The sine of \( \theta \) can be calculated as the ratio of the opposite side (height of the hill) to the hypotenuse (length of the hill). By rearranging the formula \( \sin\theta = \frac{Opposite}{Hypotenuse} \) we can solve for the Opposite side: \( Opposite = Hypotenuse * \sin\theta \)

Step 3: Plug values into the formula

Put the given values into the rearranged formula to find the Opposite side (height of the hill): \( Opposite = 45 * \sin20^\circ \)

Step 4: Calculate the height

Using a calculator, compute the value of \( 45 * \sin20^\circ \) to get the height from the ground level to the top of the hill. Round the answer to four decimal places if necessary.

Key Concepts

Sine Formula

Understanding the sine formula is a foundational aspect of right triangle trigonometry. It encapsulates the relationship between the lengths of a right-angled triangle's sides and its angles. Specifically, the sine of an angle in a right triangle expresses the ratio of the length of the side opposite the angle to the length of the hypotenuse.

To apply this, we write the sine formula as: \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \). When solving for the height in problems like the sledding hill, we rearrange the formula to \( \text{Opposite} = \text{Hypotenuse} \cdot \sin(\theta) \), substituting the known angle and hypotenuse to find the unknown side.

Trigonometric Functions

Trigonometric functions are vital tools to solve various geometry problems, especially those involving triangles. Apart from the sine function, other primary trigonometric functions include cosine and tangent. While sine relates the opposite side to the hypotenuse, the cosine deals with the adjacent side and the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side of the angle in question.

Mathematically we represent them as:\

\
- \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)\
- \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)\

These functions extend beyond right triangles to unit circles and waves, illustrating the versatility of trigonometry.

Right-Angled Triangle

A right-angled triangle is the cornerstone of trigonometry, characterized by one angle measuring exactly 90 degrees. The longest side opposite the right angle is known as the hypotenuse, and the two other sides are referred to as the adjacent and opposite, relative to the angle being considered.

This distinction between sides is crucial when applying trigonometric formulas, as knowing which side corresponds to each term in the formula is essential to find the unknown measurements. In practical problems like the hill, identifying the right-angled triangle and labeling the sides correctly are the first critical steps for a successful calculation.

Real-world Applications of Trigonometry

Trigonometry is not just an abstract mathematical concept; it has myriad real-world applications. It's used in fields such as engineering, physics, architecture, and even in the analysis of sound waves.

For example, by using trigonometry, architects can determine the correct angles and lengths of materials needed for a stable structure. In navigation, it helps in calculating distances and plotting courses, and in the digital world, it's used for computer graphics and simulations.

The sled hill problem illustrates an everyday application of trigonometry. By using the sine formula, we can measure the height of the hill, vital for safety assessments or planning construction projects in the area.