Question

Write the given expression as a product of two trigonometric functions of different frequencies. \(\cos \pi t+\cos 2 \pi t\)

Short answer

Question: Express the sum \(\cos(\pi t) + \cos(2\pi t)\) as a product of two trigonometric functions. Answer: \(\cos (\pi t) + \cos(2 \pi t) = 2 \cos(\frac{3}{2}\pi t) \cos(\frac{1}{2}\pi t)\)

Step-by-step solution

Identify the values of 'a' and 'b'

Compare the given expression with the sum-to-product formula, we need to find the values of "a" and "b" that satisfies the following equation: \(2 \cos(a) \cos(b) = \cos(\pi t) + \cos(2\pi t)\) To do so, assign each term in the expression as follows: \(\cos(\pi t) = \cos(a+b)\) and \(\cos(2\pi t) = \cos(a-b)\)

Calculate the values of 'a' and 'b'

To solve the equations in step 1, add the two equations to get: \(\pi t = a+b\) and \(2\pi t = a-b\) Now solve the system of equations for a and b Add the two equations: \(\pi t + 2\pi t = a + b + a - b\) \(3\pi t = 2a\) \(a = \frac{3}{2}\pi t\) Then, substitute this value to either equation, let's use the first equation: \(\pi t = a+b = \frac{3}{2}\pi t + b\) \(b = \frac{1}{2}\pi t\)

Apply the sum-to-product formula

Now substitute the values of 'a' and 'b' we found before into the sum-to-product formula: \(2 \cos(a) \cos(b) = 2 \cos(\frac{3}{2}\pi t) \cos(\frac{1}{2}\pi t)\) So the given expression can be written as a product of two trigonometric functions of different frequencies: \(\cos (\pi t) + \cos(2 \pi t) = 2 \cos(\frac{3}{2}\pi t) \cos(\frac{1}{2}\pi t)\)

Key concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the ratios of its sides in a right-angled triangle. They are the basis of trigonometry and can be found in various mathematical, physics, engineering, and real-world applications such as waves, oscillations, and circadian rhythms. The most commonly used trigonometric functions include sine (\text{sine}), cosine (\text{cosine}), and tangent (\text{tangent}). Interestingly, these functions also have reciprocal functions: cosecant, secant, and cotangent, respectively.

These functions can be used in a variety of ways, including the analysis of periodic phenomena. For instance, the expression \(\cos \pi t + \cos 2 \pi t\) describes the superposition of two cosine functions with different frequencies, which is often a topic in studies involving waves and vibrations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where they are defined. They are invaluable tools for simplifying and solving trigonometry equations, calculus problems, and in many areas of mathematics. Some of the most fundamental identities are the Pythagorean identity, the angle sum and difference identities, and the double angle formulas.

Particularly relevant to our exercise is the sum-to-product identity, which is used to express the sum of two trigonometric functions with different arguments in terms of products. This is exemplified by rewriting \(\cos \pi t + \cos 2 \pi t\) as \(2 \cos(\frac{3}{2}\pi t) \cos(\frac{1}{2}\pi t)\), substantially simplifying the original expression.

Frequency of Trigonometric Functions

The frequency of a trigonometric function is the rate at which the function repeats its values. It is an essential concept in oscillatory systems, such as waves and vibrations. For cosine and sine functions, frequency is related to the coefficient of the variable inside the function, which represents the angular frequency. In periodic functions such as \(\cos \pi t\), the frequency can be interpreted as the number of cycles that occur within a unit of time.

In the context of our example, \(\cos \pi t\) and \(\cos 2 \pi t\) have different frequencies: the latter oscillates twice as often as the former in the same time interval. This is observable in the differing coefficients of \(t\) within the argument of each cosine function.

System of Equations

A system of equations is a set of two or more equations with the same variables. Solving a system of equations means finding the values of the variables that make all the equations true simultaneously. In algebra, systems can be solved using various methods, including substitution, elimination, and matrix operations. These techniques are indispensable in many fields, such as economics, engineering, and physics, where multiple conditions can be described simultaneously.

For our trigonometric exercise, we used the substitution method to solve the system made by setting \(\cos(\pi t)\) equal to \(\cos(a+b)\) and \(\cos(2\pi t)\) equal to \(\cos(a-b)\). The solutions for \(a\) and \(b\) were then used to transform the sum of two cosines into a product, thereby applying the sum-to-product identity effectively.