Problem 72 - Precalculus Essentials

Question

Let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c\). $$\tan (-t)-\tan t$$

Step-by-Step Solution

Answer

The expression \(\tan (-t)-\tan t\) in terms of \(a, b, c\) is \(-2c\).

Step 1: Simplify expression using the odd-even property of tangent

Since the tangent function is an odd function, we can use the identity \(\tan(-t) = -\tan(t)\). So, the expression becomes \(-\tan(t) - \tan(t)\). This simplifies to \(-2\tan(t)\).

Step 2: Substitute Variables

Substitute the given variable \(c\) into the equation from step 1, as \(\tan(t) = c\). Thus, \(-2\tan(t)\) becomes \(-2c\).

Key Concepts

Odd Function

Odd functions have a very special and unique property. An odd function is any function where its value at \( -x \) is the negation of its value at \( x \). In mathematical terms, this is expressed as \( f(-x) = -f(x) \). This symmetry means the graph of an odd function is symmetric with respect to the origin.

Let's take a look at an example using the tangent function, which is an odd function. The identity \( \tan(-t) = -\tan(t) \) shows that if you take the tangent of a negative angle, you simply take the negative of the tangent of the original angle. This property is very useful when simplifying expressions involving odd functions because it often reduces the complexity of calculations by allowing direct substitutions or transformations.

To recap, the key aspects of odd functions are:

• They have rotational symmetry around the origin.
• Their function values at negative inputs are the negation of their function values at positive inputs.
• This property is used widely in calculus and trigonometry for simplifying expressions and solving equations.

Tangent Function

The tangent function is a fundamental trigonometric function. It can be understood as the ratio of the sine and cosine functions, expressed with the formula:\(\tan(t) = \frac{\sin(t)}{\cos(t)}.\)This relationship directly relates to the unit circle, where for an angle \( t \), the tangent corresponds to the length of the line segment that touches the circle exactly once.

The tangent function is an odd function, as we previously discussed, meaning that the identity \( \tan(-t) = -\tan(t) \) applies. This identity is crucial when manipulating expressions involving tangent. It often shows up in problems involving symmetry or when evaluating periodic functions, particularly at intervals involving \( \pi \).

Key elements of the tangent function include:

• Zero at angles where sine is zero, except at multiples of \( \frac{\pi}{2} \) where cosine becomes zero.
• A period of \( \pi \), meaning it repeats every \( \pi \) units.
• Asymptotes at angles where cosine is zero, leading to undefined tangent values.

Trigonometric Substitution

Trigonometric substitution is a powerful technique used primarily in calculus to simplify integrals that seem difficult to solve at first glance. The idea is to use trigonometric identities to transform a complicated expression into one that is easier to integrate.

This method involves substituting trigonometric functions for algebraic expressions. For example, if you are integrating a function that involves \( \sqrt{1 - x^2} \), you can use the substitution \( x = \sin(t) \), because \( \sin^2(t) + \cos^2(t) = 1 \) holds true for all angles \( t \). It transforms the integral into a trigonometric form, making it simpler to evaluate.

Here’s why trigonometric substitution is helpful:

• Transforms complex algebraic forms into manageable trigonometric integrals.
• Relies on well-known trigonometric identities.
• Often used for problems involving roots and quadratic expressions.

By using strategic trigonometric identities, substitution can clear the way for complex integration, making difficult problems approachable.