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Chapter 5: Problem 19

A ladder that is 20 feet long leans against the side of a house. The angle of elevation of the ladder is \(75^{\circ} .\) Find the height from the top of the ladder to the ground.

Short Answer

Expert verified
\[h \approx 19.4 \text{ feet}\] after rounding to the nearest decimal.

Step by step solution

01

Understand the Problem

The problem describes a right-angled triangle formed by a 20-feet long ladder leaning against a house. The ladder makes an angle of \(75^{\circ}\) with the ground. Using the sine of this angle, one can find the vertical distance from the ground to the top of the ladder.
02

Set Up the Trigonometric Relationship

Since the sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse, we can write: \(\sin(75^{\circ}) = h / 20\), where \(h\) is the height from the ground to the top of the ladder.
03

Solve for the Unknown

To find the height, we rearrange the equation to isolate \(h\): \(h = 20 \cdot \sin(75^{\circ})\). Therefore, compute \(h\) by multiplying 20 feet by the sine of 75 degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Right-Angled Triangles
Right-angled triangles are the foundation of trigonometry. In these triangles, one of the angles is exactly 90 degrees. This creates a unique scenario where we can apply trigonometric functions. The longest side is called the hypotenuse and it is opposite the right angle. The two shorter sides are referred to as the adjacent and opposite sides, depending on their relation to the angle in question.
In the context of the ladder problem, the ladder itself represents the hypotenuse, reaching from the ground to the house. The base of the ladder is the adjacent side, lying flat along the ground, and the height to the top of the ladder makes up the opposite side.
Exploring the Angle of Elevation
The angle of elevation is a crucial concept in trigonometry. It is the angle formed between the horizontal line and the line of sight from an observer's eye to an object above the horizontal. This angle helps in determining heights and distances that are otherwise difficult to measure directly.
In the ladder exercise, the angle of elevation of 75 degrees is the angle between the ground and the ladder. Knowing this angle is essential because it binds the trigonometric relationships that allow us to find the unknown height, that is, how high the top of the ladder reaches on the house.
Calculating with the Sine of an Angle
The sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This fundamental trigonometric function allows us to find other sides of the triangle when one side and an angle are known.
Referring back to the ladder question, we use the sine of the 75-degree angle because it involves the vertical distance (opposite side) and the length of the ladder (hypotenuse). By solving the equation \( \sin(75^{\circ}) = \frac{h}{20} \), we can find the height \( h \) by multiplying the hypotenuse with the sine of the angle, providing the direct vertical distance from the ground to the top of the ladder.
Grasping Trigonometric Functions
Trigonometric functions, including sine, cosine, and tangent, are essential tools in studying right-angled triangles. Each function relates to a specific angle and the ratios of the triangle's sides. They transform geometric problems into solvable mathematical equations. This allows us to calculate unknown lengths and angles efficiently.
Using the ladder example, we specifically employ the sine function to determine the vertical height. However, all trigonometric functions serve various roles in different scenarios. For instance:
  • Sine (\( \sin \)): Opposite over hypotenuse
  • Cosine (\( \cos \)): Adjacent over hypotenuse
  • Tangent (\( \tan \)): Opposite over adjacent
By understanding these functions, you can tackle other complex problems and explore the fascinating world of trigonometry with ease.

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Most popular questions from this chapter

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\sin \theta=\frac{5}{6}\)

You are given the value of tan \(\theta .\) Is it possible to find the value of \(\sec \theta\) without finding the measure of \(\theta ?\) Explain.

A six-foot person walks from the base of a streetlight directly toward the tip of the shadow cast by the streetlight. When the person is 16 feet from the streetlight and 5 feet from the tip of the streetlight's shadow, the person's shadow starts to appear beyond the streetlight's shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities and use a variable to indicate the height of the streetlight. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the streetlight?

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-270^{\circ}$$

A buoy oscillates in simple harmonic motion as waves go past. The buoy moves a total of 3.5 feet from its high point to its low point (see figure), and it returns to its high point every 10 seconds. Write an equation that describes the motion of the buoy, where the high point corresponds to the time \(t=0.\)
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