Problem 37 - Algebra and Trigonometry

Question

In Exercises \(33-42,\) let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c .\) $$ \sin (t+2 \pi)-\cos (t+4 \pi)+\tan (t+\pi) $$

Step-by-Step Solution

Answer

The expression \(\sin (t+2 \pi)-\cos (t+4 \pi)+\tan (t+\pi)\) in terms of \(a, b,\) and \(c\) is \(a - b + c\).

Step 1: Substitute the given values

Replace the given trigonometric functions with their variables \(\sin t=a, \cos t=b, \tan t=c\). The first step is to write the given expression \(\sin (t+2 \pi)-\cos (t+4 \pi)+\tan (t+\pi)\) in terms of \(a, b,\) and \(c .\)

Step 2: Remember the periodicity of the functions

The cosine function and the sine function are periodic with period \(2\pi\). This means that for any \(t\), \(\sin(t+2\pi)=\sin(t)\) and for any \(t\), \(\cos(t+2\pi)=\cos(t)\) regardless of the number of times \(2\pi\) is added to \(t\). Therefore, \(\sin(t+2\pi)=\sin t\) and \(\cos(t+4\pi)=\cos t\).

Step 3: Apply the periodicity of the tangent function

The tangent function is periodic with period \(\pi\). This means that for any \(t\), \(\tan(t+\pi)=\tan t\), regardless of the number of times \(\pi\) is added to \(t\). Therefore, we can say that \(\tan(t+\pi)=\tan t\).

Step 4: Substitute the functions with their values

Using the information from the previous steps, we can substitute \(\sin t, \cos t, \tan t\) with their corresponding variables \(a, b, c\). Thus, we get the expression \(\sin(t+2\pi) - \cos(t+4\pi) + \tan(t+\pi)\) equals to \(a - b + c\).

Key Concepts

Periodicity of Sine and Cosine

The sine and cosine functions are fundamental in trigonometry, each with a distinct pattern. A critical aspect of these functions is their periodicity. Both \(\sin(t)\) and \(\cos(t)\) have a period of \(2\pi\).
This simply means that every \(2\pi\) units along the horizontal axis, the graph of \(\sin(t)\) and \(\cos(t)\) repeats.

• For the sine function, \(\sin(t + 2\pi) = \sin(t)\) for any value of \(t\). This implies that after a full cycle, the sine function returns to its initial value.
• Similarly, for the cosine function, \(\cos(t + 2\pi) = \cos(t)\) also holds true, meaning it repeats its values every \(2\pi\) as well.

Understanding this property allows us to simplify expressions involving \(\sin(t + k\cdot 2\pi)\) or \(\cos(t + k\cdot 2\pi)\) for any integer \(k\), as they will always be equivalent to \(\sin(t)\) or \(\cos(t)\) respectively.

Periodicity of Tangent

The tangent function, \(\tan(t)\), exhibits its own pattern distinct from sine and cosine.
The period of the tangent function is \(\pi\), which is half of sine and cosine.

• This period implies that \(\tan(t + \pi) = \tan(t)\) for any value of \(t\).
• Unlike sine and cosine, which repeat every \(2\pi\), the tangent completes one full cycle in just \(\pi\) units, leading to more frequent repetitions.

This concept is crucial when solving trigonometric problems involving transformations such as \(\tan(t + k\pi)\), simplifying to \(\tan(t)\) for any integer \(k\). Thus, understanding the periodicity of \(\tan(t)\) can simplify complex trigonometric expressions.

Substitution in Trigonometric Expressions

Substitution is a valuable tool in solving trigonometric problems, especially when working with expressions involving multiple trig functions.
With substitution, expressions can be rewritten in terms of simpler variables.

• For example, if you know that \(\sin(t) = a\), \(\cos(t) = b\), and \(\tan(t) = c\), you can represent these functions in their simpler forms using \(a\), \(b\), and \(c\).
• This is particularly useful when dealing with expressions that involve transformations or shifts, such as \(\sin(t + 2\pi)\) or \(\tan(t + \pi)\), where you can apply the periodicity first, and then substitute.

Substitution allows for easier manipulation and simplification, converting intricate trigonometric expressions into algebraic ones that are more straightforward to work with.