Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\tan ^{-1}(-\sqrt{3})$$

Step-by-Step Solution

Answer

The exact value of \(y\) is \(-\frac{\pi}{3}\).

Step 1: Understand the Inverse Function

The function \(y = \tan^{-1}(-\sqrt{3})\) means we are looking for an angle \(y\) whose tangent is \(-\sqrt{3}\). The angle \(y\) should be within the range of the inverse tangent function, which is \(-\frac{\pi}{2} < y < \frac{\pi}{2}\).

Step 2: Think About Tangent Values

Recall the values of tangent for commonly known angles: \(\tan\left(\frac{\pi}{3}\right) = \sqrt{3}\) and \(\tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}\). Since \(\tan(y) = -\sqrt{3}\), the angle \(y\) likely corresponds to \(\tan\left(-\frac{\pi}{3}\right)\).

Step 3: Verify Angle in Right Quadrant

Since \(y\) is within \(-\frac{\pi}{2} < y < \frac{\pi}{2}\) and must provide a tangent of \(-\sqrt{3}\), \(y\) should be negative. Therefore, \(y = -\frac{\pi}{3}\) satisfies both the tangent value and the range.

Key Concepts

Tangent Function

The Tangent Function is part of the family of trigonometric functions, alongside sine and cosine. It is typically represented as \( \tan(\theta) \), where \( \theta \) (theta) is the angle. The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the adjacent side of the angle.

• This ratio provides a measure of the angle's steepness or slope.
• The tangent function is periodic, repeating its values every \( \pi \) radians.
• Unlike sine and cosine, which have a range between -1 and 1, the tangent function can take any real number value.

Remember, the tangent function can produce very large numbers, even for angles that are not that large, which can be useful in various mathematical applications. When it nears \( \frac{\pi}{2} \), the tangent grows towards infinity.

Trigonometric Identities

Trigonometric Identities are fundamental tools in trigonometry that relate the trigonometric functions to one another. These identities are crucial for simplifying expressions, solving equations, and proving other mathematical theorems.

• The most basic trigonometric identity is \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
• Another important identity is the ``Pythagorean identity'': \( \sin^2(x) + \cos^2(x) = 1 \), which is the foundation for many other identities.
• These identities are useful in converting complex trigonometric expressions into simpler ones.

Understanding these identities helps solve problems involving inverse trigonometric functions, where we need to find angles given some trigonometric values. Like in the exercise, we utilized the known tangent values of certain angles to determine \( y \), making such calculations manageable.

Angle Ranges

Understanding Angle Ranges is crucial when working with inverse trigonometric functions. Each function has a specific domain and range that define the possible input and resulting output.

• For \( \tan^{-1}(x) \), the angle \( y \) is typically constrained to \( -\frac{\pi}{2} < y < \frac{\pi}{2} \).
• This range helps identify the correct angle associated with a given tangent value.
• Other inverse functions, like \( \sin^{-1}(x) \) and \( \cos^{-1}(x) \), also have defined ranges, ensuring properties like one-to-one correspondence.

When solving the exercise, this range ensured that our angle \( y = -\frac{\pi}{3} \) was correctly placed, as it fit well within the allowed values, making our solution consistent with the constraints of the inverse trigonometric function.