Problem 16 - Algebra and Trigonometry

Question

Find the exact value of each expression. $$ \tan ^{-1}(-1) $$

Step-by-Step Solution

Answer

\(-\pi/4\)

Step 1: Remember Properties

The inverse tangent function, also referred to as \(\tan^{-1}\) or arctangent, undoes the tangent function. It tells us which angle has a tangent equal to a given value. The result is typically given in radians and lies in the range \(-\pi/2\) to \(\pi/2\).

Step 2: Recall Tangent Value

Looking to the unit circle, we can recall that tangent is equivalent to the ratio of the y-coordinate to the x-coordinate. This ratio is -1 at the angle \(\pi/4\) in the second quadrant, so \(\tan^{-1}(-1) = -\pi/4\).

Key Concepts

Understanding Arctangent

The arctangent, denoted as \( \tan^{-1} \) or simply arctan, is the inverse function of the tangent. It essentially reverses what the tangent function does. If the tangent of an angle gives us a certain value, the arctangent tells us which angle it was. Think of it as finding the question when we know the answer. For example, if we know that the tangent of an angle is \(-1\), the arctangent helps us find that specific angle. One crucial aspect of arctangent is that its result is typically within the range of \(-\pi/2\) to \(\pi/2\). This is an interval on the unit circle where the tangent function has an unambiguous inverse. Whenever you see \( \tan^{-1}(x) \), think: "What angle gives me \(x\) when I apply the tangent function?" Adjust your mindset to navigate angles rather than just coordinates and values.

Exploring the Unit Circle

The unit circle is a fundamental tool in trigonometry that helps us understand the behavior of sine, cosine, and tangent functions. It's a circle with a radius of one, centered at the origin of a coordinate plane. On the unit circle:

• The x-coordinate represents the cosine of an angle.
• The y-coordinate represents the sine of an angle.
• The tangent of an angle is represented by the slope \( \frac{{y}}{{x}} \).

The unit circle allows us to visualize the angle \(-\pi/4\) (or -45 degrees) where the coordinates would make the tangent \(-1\). Here, the y and x values are equal in magnitude but opposite in sign, which makes their ratio \(-1\). That's why when we see \( \tan^{-1}(-1) \), we refer to the unit circle and see that the appropriate angle is \(-\pi/4\). Visualizing these relationships makes trigonometric problems much easier to solve.

Decoding Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any angle. They play a vital role in simplifying complex trigonometric equations and understanding relationships between angles. Some core identities include:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal Identites: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
• Quotient Identity: \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)

These identities can be mixed and matched to solve various problems involving angles and are incredibly useful when dealing with inverse functions like arctangent. Knowing how to switch between forms using identities like \( \tan(45^\circ) = 1 \) can help decode expressions like \( \tan^{-1}(-1) \). Using identities simplifies the connection between geometric facts from the unit circle and algebraic manipulation, making trigonometry a breeze.