Chapter 9: Problem 4
If \(\theta\) is an acute angle, solve the equation \(\tan \theta=\frac{1}{2} .\) Express your answer in degrees, rounded to one decimal place.
Short Answer
Expert verified
\theta = 26.6°
Step by step solution
01
Understand the tangent function
The tangent of an angle \(\theta\) in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. The equation given is \(\tan \theta = \frac{1}{2}\), implying that we need to find an angle \(\theta\) whose tangent value is \(\frac{1}{2}\).
02
Use inverse tangent function
To find \(\theta\), take the inverse tangent (arctangent) of both sides of the equation: \(\theta = \tan^{-1}(\frac{1}{2})\). This will give the principal value of the angle for which the tangent is \(\frac{1}{2}\).
03
Calculate using a calculator
Use a calculator to find the arctangent value of \(\frac{1}{2}\). Make sure the calculator is in degree mode. Input: \(\tan^{-1}(\frac{1}{2})\). The calculator will provide the value of \(\theta\) in degrees.
04
Round the answer
Round the value obtained from the calculator to one decimal place for the final answer. Suppose the value obtained is approximately \[26.5651\text{...} \], round this to \(\theta \) = 26.6 degrees.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Tangent Function
The tangent function, often abbreviated as 'tan', is one of the primary trigonometric functions. It relates an angle in a right triangle to the ratio of two of its sides.
The basic formula for the tangent of an angle \theta\ is: \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\). This means that to find the tangent value of an angle, you divide the length of the side opposite the angle by the length of the side adjacent to the angle.
For example, if in a right triangle, the side opposite the angle \theta\ is 5 units long and the adjacent side is 10 units long, then \( \tan \theta = \frac{5}{10} = 0.5 \).
Understanding the tangent function is crucial when solving trigonometric equations, as it helps relate the geometric properties of triangles to angular measurements.
The basic formula for the tangent of an angle \theta\ is: \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\). This means that to find the tangent value of an angle, you divide the length of the side opposite the angle by the length of the side adjacent to the angle.
For example, if in a right triangle, the side opposite the angle \theta\ is 5 units long and the adjacent side is 10 units long, then \( \tan \theta = \frac{5}{10} = 0.5 \).
Understanding the tangent function is crucial when solving trigonometric equations, as it helps relate the geometric properties of triangles to angular measurements.
Inverse Trigonometric Functions
Inverse trigonometric functions help us find angles when we know the values of the trigonometric functions. The inverse of the tangent function is called the arctangent, denoted as \(\tan^{-1}\) or arctan.
The function \(\tan^{-1}(x)\) returns the angle \(\theta\) whose tangent is \(x\). For instance, if \(\tan \theta = \frac{1}{2}\), then \(\theta = \tan^{-1}(\frac{1}{2})\).
Using the inverse tangent function on a calculator can give you the angle in degrees or radians (ensure that your calculator is set to the correct mode). For our example, using \(\tan^{-1}(\frac{1}{2})\) yields approximately 26.6 degrees. This angle is the one that has a tangent value of \frac{1}{2}\.
The function \(\tan^{-1}(x)\) returns the angle \(\theta\) whose tangent is \(x\). For instance, if \(\tan \theta = \frac{1}{2}\), then \(\theta = \tan^{-1}(\frac{1}{2})\).
Using the inverse tangent function on a calculator can give you the angle in degrees or radians (ensure that your calculator is set to the correct mode). For our example, using \(\tan^{-1}(\frac{1}{2})\) yields approximately 26.6 degrees. This angle is the one that has a tangent value of \frac{1}{2}\.
Right Triangle Trigonometry
Right triangle trigonometry involves understanding the relationships between the angles and sides of right triangles. Each angle in a right triangle helps form ratios (sine, cosine, and tangent) based on the triangle's sides.
If we have a right triangle and the tangent of an angle \(\theta\) is known (\(\tan \theta = \frac{1}{2}\)), we can use this information to find other properties of the triangle. We often use trigonometric functions in various applications such as physics, engineering, and everyday problem-solving.
Knowing right triangle trigonometry allows us to solve problems like finding a missing side when an angle and one side are known, or like this exercise, identifying an angle when its tangent value is given.
If we have a right triangle and the tangent of an angle \(\theta\) is known (\(\tan \theta = \frac{1}{2}\)), we can use this information to find other properties of the triangle. We often use trigonometric functions in various applications such as physics, engineering, and everyday problem-solving.
Knowing right triangle trigonometry allows us to solve problems like finding a missing side when an angle and one side are known, or like this exercise, identifying an angle when its tangent value is given.
Angle Measurement
Angle measurement is a fundamental concept in trigonometry. Angles can be measured in degrees or radians. One full circle is 360 degrees or \(2\text{π}\) radians.
When solving trigonometric equations, measurements are typically in degrees unless specified otherwise. In this exercise, the angle \(\theta\) was found using the inverse tangent function and required us to round the result to one decimal place.
Using a calculator, we determined that \(\tan^{-1}(\frac{1}{2})\) is approximately 26.5651 degrees. Since we needed to round this to one decimal place, the answer is 26.6 degrees. Accurate angle measurement and understanding how to convert between degrees and radians are crucial skills in trigonometry.
Knowing how to measure angles precisely allows for accurate calculations and problem-solving in both theoretical and practical contexts.
When solving trigonometric equations, measurements are typically in degrees unless specified otherwise. In this exercise, the angle \(\theta\) was found using the inverse tangent function and required us to round the result to one decimal place.
Using a calculator, we determined that \(\tan^{-1}(\frac{1}{2})\) is approximately 26.5651 degrees. Since we needed to round this to one decimal place, the answer is 26.6 degrees. Accurate angle measurement and understanding how to convert between degrees and radians are crucial skills in trigonometry.
Knowing how to measure angles precisely allows for accurate calculations and problem-solving in both theoretical and practical contexts.
