Problem 42 - 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 \(c .\) $$ \begin{aligned} -\cos t+7 \cos (t+1000 \pi)+\tan t+\tan (t+999 \pi)+\\\ \sin t+\sin (t-1000 \pi) \end{aligned} $$

Step-by-Step Solution

Answer

The given expression in terms of \(a, b,\) and \(c\) is \(6b + 2a + 2c\).

Step 1: Identify Periodicity

Identify the periodicity nature of the trigonometric functions. The sine and cosine functions are periodic with period \(2\pi\), and the tangent function has a period of \(\pi\). So, \( \sin(t + 2n\pi) = \sin t\), \( \cos(t + 2n\pi) = \cos t\) and \( \tan(t + n\pi) = \tan t\), for any integer \(n\).

Step 2: Apply Periodicity

Apply the periodic properties to the expression. \(- \cos t + 7 \cos (t+1000 \pi) + \tan t + \tan (t+999 \pi) + \sin t + \sin (t-1000 \pi)\) simplifies to \(- \cos t + 7 \cos t + \tan t + \tan t + \sin t + \sin t\).

Step 3: Rewrite using Given Equivalents

Replace the trigonometric functions with the given equivalents. Using the given \(\sin t = a, \cos t = b, \tan t = c\), the expression becomes \(-b + 7b + c + c + a + a\). Simplifying this expression gives \(6b + 2a + 2c\).

Key Concepts

Periodicity of Trigonometric Functions

Understanding the periodicity of trigonometric functions is crucial in simplifying complex trigonometric expressions. Periodicity refers to the attribute of trigonometric functions to repeat their values at regular intervals, known as periods. For instance, the sine and cosine functions have a period of \(2\pi\), meaning that adding or subtracting multiples of \(2\pi\) does not change their value. As such, \(\sin(t + 2n\pi) = \sin t\) and \(\cos(t + 2n\pi) = \cos t\) for any integer value of \(n\).

Similarly, the tangent function repeats after every \(\pi\), so \(\tan(t + n\pi) = \tan t\). This concept is applied in the exercise to simplify the expression \( -\cos t + 7 \cos (t+1000 \pi) + \tan t + \tan (t+999 \pi) + \sin t + \sin (t-1000 \pi)\) by recognizing that adding multiples of the functions' periods results in the original functions. Therefore, knowing the periodicity helps simplify expressions by reducing them to more manageable terms.

Simplifying Trigonometric Expressions

The process of simplifying trigonometric expressions often involves recognizing patterns and applying trigonometric identities. In our exercise, after identifying the periodic properties, the complicated looking expression is simplified by using these periodic properties to transform the terms with large angle shifts (in terms of \(\pi\)) back to their basic trigonometric function form. This transformation is possible because the additional multiples of \(\pi\) do not affect the value of the sine, cosine, or tangent functions, due to their periodic nature.

Once the function has been reduced to its simplest form, further simplification can be achieved by grouping like terms. This is what was done in the final steps of the solution, which leads to the expression \(6b + 2a + 2c\) when replacing trigonometric functions with the variables \(a, b, \text{ and } c\). This approach of simplifying by using identities and grouping like terms is a powerful strategy to make complex trigonometric expressions more understandable and solvable.

Trigonometric Functions in Terms of Variables

When working with trigonometric functions in terms of variables, it can be highly beneficial to express trigonometric functions as algebraic expressions of known variables. This is especially useful when the variables represent the known values of these functions for a particular angle. In our exercise example, each trigonometric function is substituted by variables: \(\sin t = a\), \(\cos t = b\), and \(\tan t = c\). This substitution is an effective technique to transform a trigonometric expression into an algebraic one that is much simpler to handle.

By expressing trigonometric functions in terms of variables, we can perform algebraic operations just like with any other variable expressions. This often reveals solutions more clearly and allows for easier manipulation. It is a powerful algebraic tool that expands the possibilities for solving and understanding trigonometric problems while providing a bridge between trigonometry and algebra.