Problem 343 - Precalculus

Question

For the following exercises, use reference angles to evaluate the given expression. If \(\tan (t)=-0.6,\) what is the \(\tan (-t) ?\)

Step-by-Step Solution

Answer

\(\tan(-t) = 0.6\)

Step 1: Understanding the Problem

We need to find the value of \(\tan(-t)\) given that \(\tan(t) = -0.6\). The tangent function is an odd function, meaning that \(\tan(-x) = -\tan(x)\). Therefore, we can use this property to find \(\tan(-t)\).

Step 2: Applying the Odd Function Property

Since the tangent is an odd function, we directly apply the formula \(\tan(-t) = -\tan(t)\). Substituting the given value \(\tan(t) = -0.6\), we get \(\tan(-t) = -(-0.6)\).

Step 3: Simplifying the Expression

Simplify the expression \(\tan(-t) = -(-0.6)\). Double negatives cancel out, so \(\tan(-t) = 0.6\).

Key Concepts

Reference Angles

When dealing with trigonometric functions and their associated angles, reference angles help simplify calculations. A reference angle is the smallest angle that the terminal side of the given angle makes with the x-axis. This angle always measures between 0 and 90 degrees, or 0 and \(\frac{\pi}{2}\) radians. Understanding reference angles is essential because they allow you to use known values of trigonometric functions in common quadrants, reducing errors in computation. For any angle in standard position:

• If the angle is in the first quadrant, the reference angle is the angle itself.
• If it's in the second quadrant, subtract the angle from 180° or \(\pi\).
• If in the third quadrant, subtract 180° or \(\pi\) from the angle.
• In the fourth quadrant, subtract the angle from 360° or \(2\pi\).

These rules help to find the appropriate reference angle quickly and apply it to evaluate trigonometric functions effectively.

Odd Functions

Understanding odd functions is crucial in trigonometry, especially when evaluating negative angles. Odd functions have the property that applying a negative sign to the input of the function results in the negative of the function output. In formula form, this is expressed as:\[ f(-x) = -f(x) \]The tangent function, \(\tan(x)\), is an odd function. This characteristic simplifies calculations dramatically when dealing with negative angles, as seen in this exercise:

• For \(\tan(t) = -0.6\), using the odd function property provides \(\tan(-t) = -(-0.6)\).
• The double negative cancels out, resulting in a positive 0.6.

This property is particularly helpful in avoiding unnecessary recalculations by directly using the values of known angles.

Trigonometric Identities

Trigonometric identities are invaluable tools in solving trigonometric equations and simplifying expressions. An identity is an equation that holds true for all values of the involved variables. Understanding and memorizing a few key trigonometric identities can be very helpful:- Reciprocal identities (e.g., \(\tan(x) = \frac{1}{\cot(x)}\))- Pythagorean identities (e.g., \(\tan^2(x) + 1 = \sec^2(x)\))- Even and odd function identities (e.g., \(\tan(-x) = -\tan(x)\))These identities serve as the foundation for solving complex trigonometric problems. In the context of the tangent function, knowing that it is an odd function is a specific identity critical for simplifying expressions involving negative angles. Trigonometric identities offer a shorthand way to find solutions without relying heavily on calculators, making them powerful tools in a student's mathematical toolkit.