Question

The bottom of the basket of a hot-air balloon is parallel to the level ground. One taut tether line 144 feet long is attached to the center of the bottom of the basket and is anchored to the ground at an angle of \(72^{\circ},\) as shown in the figure below. Which of the following expressions gives the distance, in feet, from the center of the bottom of the basket to the ground? A. \(\frac{144}{\cos 72^{\circ}}\) B. \(\frac{144}{\sin 72^{\circ}}\) C. 144 \(\tan 72^{\circ}\) D. 144 \(\cos 72^{\circ}\) E. 144 \(\sin 72^{\circ}\)

Short answer

Answer: 144 cos 72°

Step-by-step solution

Draw a diagram

Draw a right triangle with one vertex representing the center of the bottom of the basket, the other vertex on the ground directly below the center of the bottom of the basket, and the third vertex at the anchor point on the ground where the tether line is attached. Label the angle between the tether line and ground with \(72^{\circ}\) and the tether line as 144 feet. Label the distance we want to find from the center of the bottom of the basket to the ground as h (which corresponds to the height of the triangle).

Set up trigonometric equation

Since the angle is given between the tether line (the hypotenuse) and the ground, we can use cosine to relate the angle, the hypotenuse, and the adjacent leg which is h. Using the cosine formula: \(\cos 72^{\circ} = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)

Solve for h

Plug the given values into the equation: \(\cos 72^{\circ} = \frac{h}{144}\) Now, solve for h: \(h = 144 \cdot \cos 72^{\circ}\) Thus, the answer is \(\boxed{\text{(D) } 144 \cos 72^{\circ}}\).

Key concepts

Cosine Formula

Understanding the cosine formula is fundamental when dealing with right triangles, especially in trigonometry-focused ACT math problems. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

In mathematical terms, this is expressed as:
\[\begin{equation}\cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\end{equation}\]
where \( \theta \) is the angle in question. For instance, if you’re given the hypotenuse and an angle, you can find the length of the side adjacent to the angle by rearranging the formula:
\[\begin{equation}\text{Adjacent side} = \text{Hypotenuse} \times \cos(\theta)\end{equation}\]
This powerful tool allows you to solve many geometry problems, and it's particularly useful when you don’t have direct measurements of all sides of a triangle.

Right Triangle

A right triangle, also known as a right-angled triangle, has one angle exactly equal to \(90^\circ\). It’s the star of the show in trigonometry, forming the basis of many trigonometric definitions and properties.

In any right triangle, the side opposite the right angle is called the hypotenuse, while the other two sides are known as the legs. The legs can be referred to as 'adjacent' or 'opposite' depending on which acute angle is being considered.

The fact that one angle is fixed at \(90^\circ\) allows for predictable relationships between the angles and sides, leading to specific trigonometric ratios like sine, cosine, and tangent. These ratios help us to solve problems involving right triangles, even if some side lengths or angles are unknown.

ACT Math Problems

When tackling trigonometry on the ACT Math section, it’s common to encounter problems involving right triangles and trigonometric equations. The test assesses a variety of skills, from understanding trigonometric functions and their properties to applying formulas for problem-solving.

To succeed with these problems, you'll want to be familiar with sine, cosine, and tangent ratios, as well as the Pythagorean theorem, which relates the lengths of sides in a right triangle. Moreover, you'll need to be comfortable with manipulating equations to solve for missing values, as illustrated in the provided exercise.

Efficiently translating word problems to mathematical equations and visualizing the relevant triangles can save precious time during the test. As shown in the exercise, a step-by-step approach to first depict the problem and then apply the trigonometric equations is key to finding the correct solution.

Trigonometric Equations

Trigonometric equations are mathematical statements that involve trigonometric functions of an unknown angle. They are essential for describing the relationships in circular and periodic phenomena, as well as right triangle problems.

Solving trigonometric equations often involves isolating the variable, which is the angle, and then finding its values, typically using inverses of trigonometric functions. However, because they have periodic solutions, there might be multiple angles that satisfy a given equation. Careful attention must be paid to the domain of the problem to ensure that all possible solutions are considered.

In the context of the ACT Math test or similar, the domain is usually implied by the physical constraints of the problem, as in the case of a tether line forming a triangle on the ground, which limits the range of possible angles and thus the solutions.