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Chapter 4: Problem 42

$$\text {use a calculator to find the value of the acute}\text { angle } \theta \text { in radians, rounded to three decimal places.}$$ $$\tan \theta=0.5117$$

Short Answer

Expert verified
The value of \( \theta \) in radians is approximately the result obtained from the calculator after performing 'tan^-1(0.5117)'.

Step by step solution

01

Identify the problem

The problem provides the tangent of an angle \( \theta \) and asks to find the value of \( \theta \) in radians. The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, we know \(\tan \theta=0.5117\), so one can solve for the angle \( \theta \) using the inverse tangent function, also known as arctan or tan^-1.
02

Use the inverse tangent function

The inverse tangent of 0.5117 can be found using a calculator set to radian mode. Most calculators have a function usually denoted as 'tan^-1' or 'atan' or 'arctan'. Use this function to find \( \theta \) by entering 'tan^-1(0.5117)'.
03

Perform the calculation

After entering the inverse tangent of 0.5117 into the calculator, the result will give the measurement of angle \( \theta \) in radians, rounded to three decimal places.

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Key Concepts

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

Inverse Tangent Function
When you encounter a trigonometry problem involving the tangent of an angle, such as \( \tan \theta = 0.5117 \), the next step to find the angle itself is often to use the inverse tangent function. This function, also known as arctangent or \( \tan^{-1} \) , allows us to work backward from a known tangent value to determine the angle that produced it.

In a calculator, this function is typically labeled as \( \tan^{-1} \) or sometimes 'atan' or 'arctan'. For our exercise, setting the calculator to radian mode before using the inverse tangent function is crucial, as we want the answer in radians, not degrees. Furthermore, remember that the inverse tangent function will return the principal value, which for the tangent function typically lies between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), meaning it will yield an acute angle, in line with the requirement of our problem.
Calculating Angles in Radians
Angles can be measured in degrees or radians, and understanding the radian measure is essential in higher mathematics and scientific applications. The exercise asks us to express the angle in radians. One radian is the angle created when the arc length of a circle is equal to the circle's radius. There are \( 2\pi \) radians in a full circle, which corresponds to 360 degrees.

Therefore, to convert an angle from degrees to radians, you would multiply by \( \frac{\pi}{180} \). Conversely, to convert from radians to degrees, you would multiply by \( \frac{180}{\pi} \). For the problem at hand, since we are using the inverse tangent function, the output will be in radians by default, so no conversion is necessary once the calculator is set correctly.
Trigonometric Ratios
Trigonometric ratios provide the relationship between the angles and sides of a right triangle. There are six fundamental ratios: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each ratio is a function of an angle, often denoted as \( \theta \).

In our example, we focused on the tangent, which is the ratio of the length of the opposite side to the length of the adjacent side with respect to the angle \( \theta \). For a given angle \( \theta \), \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Knowing the value of any of these ratios gives us the means to calculate the corresponding angle through their inverse functions, as demonstrated with the inverse tangent function or \( \tan^{-1} \).

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Most popular questions from this chapter

The following figure shows the depth of water at the end of a boat dock. The depth is 6 feet at low tide and 12 feet at high tide. On a certain day, low tide occurs at 6 A.M. and high tide at noon. If \(y\) represents the depth of the water \(x\) hours after midnight, use a cosine function of the form \(y=A \cos B x+D\) to model the water's depth.

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing one complete cycle of \(y=A \sin (B x-C)\) I find it easiest to begin my graph on the \(x\) -axis.

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