Problem 67 - Contemporary Precalculus

Question

A rocket is fired straight up. The line of sight from an observer 4 miles away makes an angle of \(t\) radians with the horizontal. (a) Express \(t\) as a function of the height \(h\) of the rocket. (b) Find \(t\) when the rocket is .25 mile, 1 mile, and 2 miles high respectively. (c) When \(t=.4\) radian, how high is the rocket? (GRAPH CANNOT COPY)

Step-by-Step Solution

Answer

Question: Use the tangent function to express the angle \(t\) as a function of the rocket height \(h\). Then find the angle \(t\) in radians when: (a) \(h = 0.25\) mile, (b) \(h = 1\) mile, and (c) \(h = 2\) miles. Finally, find the height of the rocket when \(t = 0.4\) radians. Answer: The angle \(t\) can be expressed as a function of the rocket height \(h\) using the formula: \(t = \arctan{(\frac{h}{4})}\). For the given heights, we find: (a) \(t \approx 0.0624\) radians, (b) \(t \approx 0.245\) radians, and (c) \(t \approx 0.463\) radians. When \(t = 0.4\) radians, the height of the rocket is approximately \(1.53\) miles.

Step 1: Expressing t as a function of h

In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. In this problem, the tangent of the angle \(t\) is equal to the ratio of the height \(h\) to the horizontal distance, which is 4 miles. So we have: \(\tan{t} = \frac{h}{4}\) To express \(t\) as a function of \(h\), we now take the inverse tangent of both sides: \(t = \arctan{(\frac{h}{4})}\)

Step 2: Finding t for given heights

Now we will find the angle \(t\) for the given heights of the rocket: (a) For \(h = 0.25\) mile: \(t = \arctan{(\frac{0.25}{4})} \approx 0.0624\) radians (b) For \(h = 1\) mile: \(t = \arctan{(\frac{1}{4})} \approx 0.245\) radians (c) For \(h = 2\) miles: \(t = \arctan{(\frac{2}{4})} \approx 0.463\) radians

Step 3: Finding the height for given angle t

For part (c) of the problem, we are given the angle \(t=0.4\) radians and need to find the height \(h\). We can use the same formula that we derived in step 1: \(t = \arctan{(\frac{h}{4})}\) To solve for \(h\), we first take the tangent of both sides: \(\tan{t} = \frac{h}{4}\) Then, we plug in the value of \(t=0.4\) and solve for \(h\): \(\tan{0.4} = \frac{h}{4}\) \(h \approx 4 \cdot \tan{0.4} \approx 1.53\) miles. Thus, when \(t=0.4\) radians, the rocket is approximately \(1.53\) miles high.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in trigonometry, allowing us to determine the angle when the values of trigonometric ratios are known. These functions are particularly useful when it comes to solving real-world problems where you're given a ratio and need the corresponding angle.

• The inverse sine function, denoted as \( \arcsin \), finds an angle whose sine is a specific value.
• The inverse cosine function, denoted as \( \arccos \), finds an angle with a given cosine value.
• The inverse tangent function, \( \arctan \), is what we use in this exercise. It finds an angle when the tangent ratio is known.

For our rocket problem, we needed to express \( t \) in terms of \( h \), resulting in: \( t = \arctan(\frac{h}{4}) \). This formula comes from taking the inverse tangent of both sides of the equation \( \tan(t) = \frac{h}{4} \). The inverse tangent function is particularly useful when studying problems involving right triangles and their angles.

Tangent Function

The tangent function is one of the primary trigonometric functions that relate the angles of a right triangle to the ratios of two of its sides. In right-angle trigonometry, the tangent of an angle is the ratio of the length of the opposite side to the adjacent side. It can be defined as:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

In our specific problem, for the right triangle formed by the observer, the line of sight, and the rocket, the rocket's height corresponds to the opposite side, and the distance from the observer to the rocket's base, 4 miles, is the adjacent side. This gives us the equation:

\( \tan(t) = \frac{h}{4} \)
The tangent function is particularly sensitive to changes in the opposite to adjacent ratio, meaning small changes in the angle will lead to relatively larger changes in the tangent value. This characteristic is key when solving for the rocket's height based on a given angle.

Right Triangle

A right triangle is a triangle where one of its angles is exactly \( 90^\circ \) or a right angle. Right triangles simplify many calculations in trigonometry because the primary trigonometric functions (sine, cosine, and tangent) have clear, definable relationships with the triangle's sides.
The problem at hand involves such a triangle, formed by the observer, the base of the rocket's flight path, and the rocket itself. Here’s what you should know about right triangles:

• The side opposite the right angle is always the longest side, termed the hypotenuse.
• The other two sides are the legs of the right triangle. In this exercise, the height \( h \) of the rocket is one leg, and the distance from the observer to the launch site (4 miles) is the other leg.

Using these sides, trigonometric functions can be applied to calculate unknown angles or side lengths. These calculations are vital for navigating problems involving angles and distances, such as the height determination of an object like a rocket from a ground-based observer.