In this explainer, we will learn how to use Euler’s formula to prove trigonometric identities like double angle and half angle.
When we first learn about trigonometric functions and the exponential functions, they seem to have little, to nothing, in common. Trigonometric functions are periodic, and, in the case of sine and cosine, are bounded above and below by 1 and , whereas the exponential function is nonperiodic and has no upper bound. However, Euler’s formula demonstrates that, through the introduction of complex numbers, these seemingly unconnected ideas are, in fact, intimately connected and, in many ways, are like two sides of a coin.
Definition: Euler’s Formula
Euler’s formula states that for any real number ,
This formula is alternatively referred to as Euler’s relation.
Euler’s formula has applications in many area of mathematics, such as functional analysis, differential equations, and Fourier analysis. Furthermore, its applications extend into physics and engineering in such diverse areas as signal processing, electrical engineering, and quantum mechanics. In this explainer, we will focus on the its applications in trigonometry—in particular the derivations of trigonometric identities.
In the first example, we will demonstrate how we can derive trigonometric identities by using one of the properties of the exponential function and then applying Euler’s formula.
Example 1: Using Euler’s Formula to Derive Trigonometric Identities
- Use Euler’s formula to express in terms of sine and cosine.
- Given that , what trigonometric identity can be derived by expanding the exponentials in terms of trigonometric functions?
Answer
Part 1
Rewriting we can apply Euler’s formula to get
Using the odd/even identities for sine and cosine, we have
Part 2
Using Euler’s identity and our answer from part 1, we can rewrite
Expanding the parentheses, we have
Finally, we can simplify this to get
Hence, given that , we have proven the trigonometric identity .
As we saw in the last example, by using the properties of the exponential function and applying Euler’s formula, we can derive trigonometric identities. The next example will demonstrate the versatility of this method.
Example 2: Trigonometric Identities from Euler’s Formula
What trigonometric identities can be derived by applying Euler’s identity to ?
Answer
The first thing we need to do is apply one of the properties of the exponential function. The obvious choice here is the multiplicative property:
Now that we have two different but equal expressions, we can apply Euler’s formula to each one to get
By expanding the parentheses on the right-hand side, we have
Gathering together the like terms on the right-hand side, we find that
Finally, we can equate the real and imaginary parts to get the following two trigonometric identities:
Example 3: Double Angle Formulas from Euler’s Formula
Use Euler’s formula to derive a formula for and in terms of and .
Answer
The first thing we need to consider is what property of the exponential function we can apply to get two different but equal expressions. Since we are looking for identities in , it makes sense that we would consider the expression . At this point, there are two obvious choices for properties of the exponential function to use, either the power rule or the multiplicative rule. In both of these cases, we will get the same result:
Applying Euler’s formula to both of these expression, we have
By expanding the parentheses, we have
Finally, we can equate the real and imaginary parts to get the following two trigonometric identities:
Using a similar method, we can derive multiple angle formulas for and for larger values of . However, to simplify the derivation, we can use the binomial theorem.
The Binomial Theorem
For any integer , where . Sometimes, is denoted or .
In the next example, we will demonstrate how we can use the binomial theorem to simplify the derivation of multiple angle formulas for sine and cosine.
Example 4: Multiple Angle Formulas from Euler’s Formula
- Use Euler’s formula to derive a formula for and in terms of and .
- Hence, express in terms of .
Answer
Part 1
Using the properties of the exponential function, we can write
Using Euler’s formula, we can rewrite this as
Using the binomial theorem, we can expand the parentheses as follows:
Substituting in the vales of and simplifying, we have
Evaluating the powers of , we get
By equating the real and imaginary parts, we arrive at the following two identities:
Part 2
Using the identities from part 1, we can write
To express this in terms of , we need to divide both the numerator and denominator by , which results in
One of the interesting applications of Euler’s formula is that we can derive an expression for and in terms of and . In the first example, we saw that we can express in terms of sine and cosine as
Adding this to Euler’s formula gives
Hence, by dividing by 2, we derive the following formula for cosine:
Similarly, we can derive a formula for sine by considering the difference between equation (1) and Euler’s formula as follows:
Dividing by , we arrive at
These two formulas are certainly interesting in their own right. However, for the purpose of this explainer, we will demonstrate how they can be used to derive further trigonometric identities.
Example 5: Product-to-Sum Formulas from Euler’s Formula
- Derive two trigonometric identities by considering the real and imaginary parts of .
- Similarly, what two trigonometric identities can be derived by considering the real and imaginary parts of ?
Answer
Part 1
We begin by applying the multiplicative property of exponentials to rewrite
Factoring out the common factor of , we have
Using the definition of cosine in terms of the exponential function, we can rewrite this as
We can now use Euler’s formula to rewrite this as
Grouping together the real and imaginary parts on both side of the equations yields
By equating the real and imaginary parts, we get the following two trigonometric identities:
Part 2
Using a similar method, we can consider the expression . Applying the properties of the exponential function, we can rewrite this as
By factoring out , we can express this as
Using the definition of sine in terms of the exponential function, we can rewrite this as
We can now apply Euler’s formula to both sides of the equation to get
Grouping together the real and imaginary parts on both side of the equations yields
By equating the real and imaginary parts, we get the following two trigonometric identities:
In this final example, we use the definitions of sine and cosine in terms of the complex exponential function to express powers of sine and cosine in terms of multiple angle formulae.
Example 6: Powers in terms of Multiple Angles from Euler’s Formula
- Use Euler’s formula to express in the form , where , , and are constants to be found.
- Hence, find the solutions of in the interval . Give your answer in exact form.
Answer
Part 1
Using the definitions of sine and cosine in terms of the complex exponential, we can rewrite the expression as
Hence,
Using the binomial theorem, we can expand each of the sets of parentheses as follows:
We can now expand the two parentheses to get
Gathering the terms with terms results in
Hence,
We can now use the definition of sine in terms of the complex exponential to rewrite this as
Part 2
Using our answer from part 1, we can see that
Hence, finding the solutions to is equivalent to finding the solutions to
We can factor this equation as follows:
This has solutions if either or . Considering the first case, given that , when .
We can now consider the case where . Using the double angle formula for sine, we can rewrite this as
Hence,
Taking the square root of both sides of the equation, we get
Starting with the positive square root for in the range , when or . Similarly, for the negative square root, when or .
Hence, the solutions of for in the range are
Key Points
- Using Euler’s formula in conjunction with the properties of the exponential function, we can derive many trigonometric identities such as the Pythagorean identity, sum and difference formulas, and multiple angle formulas.
- We can use Euler’s theorem to express sine and cosine in terms of the complex exponential function as Using these formulas, we can derive further trigonometric identities, such as the sum to product formulas and formulas for expressing powers of sine and cosine and products of the two in terms of multiple angles.
- We can use the identities derived from Euler’s formula to help us simplify integrals and solve trigonometric equations.