Problem 101 - A Graphical Approach to Precalculus with Limits

Question

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\tan \theta=\sqrt{3}$$

Step-by-Step Solution

Answer

\( \theta = \tan^{-1}(\sqrt{3}) = 60^\circ \).

Step 1: Identify the Trigonometric Problem

We are given the equation \( \tan \theta = \sqrt{3} \) and are required to find the acute angle \( \theta \). The target is to express \( \theta \) using an inverse trigonometric function, and then also convert this angle measurement to degrees.

Step 2: Use Inverse Tangent Function

To find \( \theta \), we use the inverse function: \( \theta = \tan^{-1}(\sqrt{3}) \). This expression denotes the acute angle that produces a tangent value of \( \sqrt{3} \).

Step 3: Recognize Special Angles

Knowing the trigonometric function values for common angles, we recognize that \( \tan(60^\circ) = \sqrt{3} \). Therefore, the acute angle \( \theta \) is \( 60^\circ \).

Step 4: Convert the Angle to Degrees

Since the problem requires expressing the answer also in degrees, we conclude that the measure of \( \theta \) is \( 60^\circ \), which matches the value of \( \tan(60^\circ) \).

Key Concepts

Trigonometric Problem Solving

Trigonometric problem solving involves identifying and using relationships between angles and sides in triangles. A common type of problem includes finding an angle when given a trigonometric function's value, as in this exercise. Recognize that each trigonometric function like sine, cosine, and tangent links an angle in a right triangle to ratios of two of its sides. - For tangent, often represented as \( \tan \theta \), this is the ratio of the opposite side to the adjacent side.When a problem provides a trigonometric function's value, your task is to determine the corresponding angle that generates that specific value. These types of problems usually require a strong understanding of relationships and properties within triangles, as well as transformations involving these functions.

Inverse Tangent Function

The inverse tangent function, written as \( \tan^{-1}(x) \), allows us to find the angle whose tangent is \( x \). Given that tangent shares a cyclical pattern, inverse tangent helps pinpoint the specific angle for which the tangent equals a particular value within its principal range.- In our problem, we need to find \( \theta \) such that \( \tan(\theta) = \sqrt{3} \).- We express this as \( \theta = \tan^{-1}(\sqrt{3}) \).This indicates the exact angle which, when the tangent is taken, results in \( \sqrt{3} \). The inverse tangent function effectively "undoes" the tangent, isolating \( \theta \). Remember, the range for \( \tan^{-1}(x) \) is usually \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), making it very important in problems asking for an acute angle.

Special Angles in Trigonometry

Special angles in trigonometry refer to commonly used angles whose trigonometric function values are frequently memorized due to their regular appearance and usefulness. These include angles like 30°, 45°, and 60°. Recognizing these values is crucial, as it greatly simplifies calculations and avoids repeated function evaluations.- In the given problem, we needed to identify which angle has a tangent value of \( \sqrt{3} \).- Knowing the value \( \tan(60^\circ) = \sqrt{3} \), we easily find \( \theta = 60^\circ \).Being familiar with these angles is incredibly beneficial for quickly solving trigonometry problems without resorting to calculators or complicated computations. Remembering these key trigonomic values saves time and allows for quick problem-solving in certain trigonometry challenges.