Problem 70
Question
In this set of exercises, you will use inverse trigonometric functions to study real-world problems. Round all answers to four decimal places. The pitch of a roof is its slope, which is given as \(\frac{\text { rise }}{\text { run }}\). If the pitch of a roof is \(\frac{2}{5},\) what acute angle does it make with the horizontal? Express your answer in radians.
Step-by-Step Solution
Verified Answer
The acute angle that the roof makes with the horizontal is approximately \(\text{arctan}\left(\frac{2}{5}\right)\) radians.
1Step 1: Identify the Known Values
The known value in this exercise is the pitch of the roof, or slope, which is given as \(\frac{2}{5}\). This is the tangent of the desired angle.
2Step 2: Apply the Inverse Tangent Function
To find the acute angle, take the inverse tangent or arctan of the pitch value. This can generally be done on a scientific calculator as \(\text{arctan}\left(\frac{2}{5}\right)\).
3Step 3: Round the Radians Value
After using the inverse tangent function, round the result to four decimal places to get the final radians value.
Key Concepts
Inverse TangentPitch of a RoofTrigonometric ApplicationsRadians
Inverse Tangent
When we encounter a scenario that involves finding an angle when we know the ratio of the sides of a right triangle, the inverse tangent function comes into play. In such cases, if we have the values for the opposite side and adjacent side of the right triangle, we can use the inverse tangent function to find the measure of the angle.
The function, often denoted as \text{arctan}(x) or \text{tan}^{-1}(x), gives us the angle whose tangent is the number 'x'. Remember to have your calculator set to the correct mode—degrees or radians—based on the requirement of the problem you are solving. When the problem asks for the angle in radians, make sure your calculator is in radian mode.
For the mathematical formula, this is represented as: \( \theta = \text{arctan}(x) \), where \(\theta \) is the angle in radians and 'x' is the tangent of the angle.
The function, often denoted as \text{arctan}(x) or \text{tan}^{-1}(x), gives us the angle whose tangent is the number 'x'. Remember to have your calculator set to the correct mode—degrees or radians—based on the requirement of the problem you are solving. When the problem asks for the angle in radians, make sure your calculator is in radian mode.
For the mathematical formula, this is represented as: \( \theta = \text{arctan}(x) \), where \(\theta \) is the angle in radians and 'x' is the tangent of the angle.
Pitch of a Roof
The term 'pitch' of a roof refers to the steepness or the incline of a roof, which is of significant importance in architecture and construction. It is a measure of how high the roof rises for every horizontal unit of run. The pitch is expressed as a fraction or a ratio where the numerator represents the rise (vertical height) and the denominator represents the run (horizontal length).
The pitch of a roof plays a crucial role in shedding rain and snow, and can also impact the aesthetic appeal of a building. The larger the pitch, the steeper the roof. In practical applications when we're given the pitch, like a ratio of \(\frac{2}{5}\), we can interpret it as a rise of 2 units for every 5 units of horizontal distance.
The pitch of a roof plays a crucial role in shedding rain and snow, and can also impact the aesthetic appeal of a building. The larger the pitch, the steeper the roof. In practical applications when we're given the pitch, like a ratio of \(\frac{2}{5}\), we can interpret it as a rise of 2 units for every 5 units of horizontal distance.
Trigonometric Applications
Trigonometry is not just a subject to be studied in the classroom; it has a plethora of applications in the real world. From architecture to engineering, from astronomy to navigation, the principles of trigonometry are applied to calculate heights, distances, and angle measures.
One common application is determining the angle of elevation or depression. In navigation, trigonometry is used to set a course and maintain a good understanding of wind directions. Architects use it to design buildings and understand the physics of their structures. Trigonometry even stretches into the field of physics, where it's used to solve problems involving periodic motion like waves.
One common application is determining the angle of elevation or depression. In navigation, trigonometry is used to set a course and maintain a good understanding of wind directions. Architects use it to design buildings and understand the physics of their structures. Trigonometry even stretches into the field of physics, where it's used to solve problems involving periodic motion like waves.
Radians
In mathematics, angles can be measured in degrees or radians. A radian is the standard unit of angular measure used in many areas of mathematics. The radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
One complete revolution around a circle is equal to \(2\pi\) radians, which is equivalent to 360 degrees. Since the exercise requires the answer in radians, after finding the angle with the inverse tangent function, we express our answer in terms of \(\pi\). This provides a more accurate representation of the angle for use in mathematical and scientific calculations. After calculating, rounding the value to four decimal places ensures precision in the answer.
One complete revolution around a circle is equal to \(2\pi\) radians, which is equivalent to 360 degrees. Since the exercise requires the answer in radians, after finding the angle with the inverse tangent function, we express our answer in terms of \(\pi\). This provides a more accurate representation of the angle for use in mathematical and scientific calculations. After calculating, rounding the value to four decimal places ensures precision in the answer.
Other exercises in this chapter
Problem 69
Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$25^{\circ} 45^{\prime} 15^{\prime \prime}$$
View solution Problem 69
In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal place
View solution Problem 70
Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$32^{\circ} 30^{\prime} 30^{\prime \prime}$$
View solution Problem 70
In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal place
View solution