Problem 16
Question
A pole standing on level ground makes an angle of \(85.8^{\circ}\) with the horizontal. The pole is supported by a 22.0 -ft prop whose base is \(12.5 \mathrm{ft}\) from the base of the pole. Find the angle made by the prop with the horizontal.
Step-by-Step Solution
Verified Answer
The angle made by the prop with the horizontal is 4.2 degrees.
1Step 1: Understanding the Scenario
Visualize the situation as a right triangle where the pole is one side perpendicular to the ground, the prop is the hypotenuse making an angle with the ground, and the distance from the prop's base to the pole is the base of the triangle. The angle between the pole and the ground is complementary to the angle between the prop and the ground because they add up to 90 degrees.
2Step 2: Calculate the Prop's Angle with the Horizontal
Since the pole makes an angle of 85.8 degrees with the horizontal, the angle between the prop and the horizontal is the complement to this angle, i.e., 90 degrees minus 85.8 degrees. Calculate this using the equation \( \theta_{prop} = 90^\circ - \theta_{pole} \).
3Step 3: Performing the Calculation
Substitute the given angle made by the pole with the horizontal into the equation to find the angle made by the prop with the horizontal: \( \theta_{prop} = 90^\circ - 85.8^\circ = 4.2^\circ \).
Key Concepts
Understanding Trigonometry in Right Triangle ScenariosComplementary Angles in ActionNavigating Right Angle Geometry
Understanding Trigonometry in Right Triangle Scenarios
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, especially right-angled triangles. In the context of our exercise, understanding trigonometry is crucial to solving real-world problems involving right triangles.
At its core, trigonometry is about defining and really getting to know the trigonometric functions: sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides. In a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side.
These relationships allow us to solve for unknown angles and sides. In our textbook exercise, although trigonometric functions are not directly used, the foundational understanding of angles and their complements helps us determine the missing angle using simple subtraction. This principle is also critical when one needs to use trigonometric functions to solve for sides and angles in more complex situations.
At its core, trigonometry is about defining and really getting to know the trigonometric functions: sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides. In a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side.
These relationships allow us to solve for unknown angles and sides. In our textbook exercise, although trigonometric functions are not directly used, the foundational understanding of angles and their complements helps us determine the missing angle using simple subtraction. This principle is also critical when one needs to use trigonometric functions to solve for sides and angles in more complex situations.
Complementary Angles in Action
The concept of complementary angles is simple: two angles are complementary if their sum equals 90 degrees. This is especially relevant in right triangle geometry because one of the angles is always 90 degrees, making the other two angles complementary to each other.
In our pole and prop scenario, once we know the angle the pole makes with the ground, we instantly know that the angle the prop makes with the ground must be the complement to this angle so that they add up to 90 degrees. The formula \( \theta_{prop} = 90^\circ - \theta_{pole} \) makes use of this very concept and ties back to the essence of trigonometry in right triangles.
Understanding complementary angles is also essential in trigonometric identities, where functions of complementary angles are related, such as \( \sin(90^\circ - \theta) = \cos(\theta) \) and vice versa. These identities are fundamental tools in solving a myriad of trigonometry problems.
In our pole and prop scenario, once we know the angle the pole makes with the ground, we instantly know that the angle the prop makes with the ground must be the complement to this angle so that they add up to 90 degrees. The formula \( \theta_{prop} = 90^\circ - \theta_{pole} \) makes use of this very concept and ties back to the essence of trigonometry in right triangles.
Understanding complementary angles is also essential in trigonometric identities, where functions of complementary angles are related, such as \( \sin(90^\circ - \theta) = \cos(\theta) \) and vice versa. These identities are fundamental tools in solving a myriad of trigonometry problems.
Navigating Right Angle Geometry
Right angle geometry revolves around the special properties of right triangles. This field of study is critical for tasks ranging from simple constructions to advanced engineering problems.
A right triangle, by definition, has one angle that is exactly 90 degrees. This property not only defines the shape but also enables us to apply Pythagoras' theorem, where the square of the hypotenuse is equal to the sum of the squares of the other two sides \((a^2 + b^2 = c^2)\).
In our example, if we were tasked with finding the length of the prop or the pole, we could apply this theorem, knowing the lengths of the other two sides. Understanding the principles of right angle geometry underpins much of the problem-solving in both trigonometry and many practical applications like construction, navigation, and even computer graphics.
A right triangle, by definition, has one angle that is exactly 90 degrees. This property not only defines the shape but also enables us to apply Pythagoras' theorem, where the square of the hypotenuse is equal to the sum of the squares of the other two sides \((a^2 + b^2 = c^2)\).
In our example, if we were tasked with finding the length of the prop or the pole, we could apply this theorem, knowing the lengths of the other two sides. Understanding the principles of right angle geometry underpins much of the problem-solving in both trigonometry and many practical applications like construction, navigation, and even computer graphics.
Other exercises in this chapter
Problem 15
Trigonometric Functions of Any Angle by Calculator. Write, to four significant digits, the sine, cosine, and tangent of each angle. $$114^{\circ} 23^{\prime}$$
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Forces of 675 lb and 828 lb act on a body. The smaller force acts due north: the larger force acts \(\mathrm{N} 52.3^{\circ} \mathrm{E}\). Find the direction an
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State whether the following expressions are positive or negative. Do not use your calculator, and try not to refer to your book. $$\sin 174^{\circ}$$
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Trigonometric Functions of Any Angle by Calculator. Write, to four significant digits, the sine, cosine, and tangent of each angle. $$-11^{\circ} 18^{\prime}$$
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