Problem 15 - Algebra 2 and Trigonometry

Question

In \(13-24,\) find, to the nearest ten-thousandth, the radian measure \(\theta\) of a first-quadrant angle with the given function value. $$ \tan \theta=3.726 $$

Step-by-Step Solution

Answer

The radian measure is approximately 1.3014.

Step 1: Identify the Quadrant and Use Arc Function

Since the angle is in the first quadrant and the tangent function is positive in this quadrant, we use the inverse tangent function to find the angle. The formula is given by \( \theta = \tan^{-1}(3.726) \).

Step 2: Calculate the Radian Measure

Use a calculator to compute \( \tan^{-1}(3.726) \). This computation results in \( \theta \approx 1.3014 \) radians.

Step 3: Round to the Nearest Ten-Thousandth

Finally, round the result to four decimal places to get the radian measure of \( \theta \) to the nearest ten-thousandth: \( 1.3014 \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find an angle when the value of the trigonometric function is known. These functions are crucial in solving equations where the angle is the unknown variable. For example, if you know the value of the tangent function, you can find the angle by using the inverse tangent function, denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \). This function undoes what the tangent function does, which is finding the ratio of the opposite side to the adjacent side in a right triangle. The domain of inverse trigonometric functions is usually restricted to make them functions. For the inverse tangent, \( \tan^{-1}(x) \), the range is \((-\pi/2, \pi/2)\), which covers the angles in the first and fourth quadrants. This restriction ensures that each input (a ratio) has exactly one output (an angle). Hence, when we find \( \tan^{-1}(3.726) \), we are determining the angle measure in these restricted domains.

Tangent Function

The tangent function is a fundamental trigonometric function, often remembered by the acronym TOA from the mnemonic SOHCAHTOA. In a right triangle, the tangent of an angle \( \theta \) is the ratio of the length of the opposite side to the length of the adjacent side. This can be written as:\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]Unlike sine and cosine functions, tangent can take any real value, leading to a range of \((-\infty, \infty)\). Notice also that it's periodic with a period of \(\pi\), meaning the function repeats its values every \(180^\circ\) or \(\pi\) radians. The tangent function has vertical asymptotes where it is undefined, occurring at \(\frac{\pi}{2} + n\pi\) for integer \(n\). In the context of the problem, the value \(\tan \theta = 3.726\) corresponds to a first-quadrant angle, which is critical because the function’s positivity in this quadrant dictates usage of \( \theta = \tan^{-1}(3.726) \).

Radian Measure

Radian measure is a way of expressing angles using the radius of a circle. It considers how far along the arc of the circle a particular angle covers, representing angles in terms of \(\pi\). Unlike degrees, which divide a circle into 360 parts, radians use the relationship between a circle's circumference and its radius. For instance, a full circle is \(2\pi\) radians or 360 degrees, meaning \(\pi\) radians equate to 180 degrees. When solving trigonometry problems like the one with \(\tan \theta = 3.726\), it's often necessary to convert angles to radians or compute them directly in radian measure. This approach is standard in higher mathematics due to its natural relation to circular motion and calculus.In the solution, calculating \( \tan^{-1}(3.726) \) on a calculator results in a radian measure, specifically \( \theta \approx 1.3014 \) rad. It's good practice to round to four decimal places for precision, especially in academic problems. Using radians ensures consistency across different mathematical contexts, providing a universal way to deal with angles.