# Trigonometric identities and complex numbers

• PedroB
In summary, the problem involves using complex numbers to show the trigonometric identity sin(x)+cos(x)=(√2)cos(x-∏/4). The solution requires rewriting the expression in terms of complex exponentials and using identities such as cosx = Re(eix).
PedroB

## Homework Statement

Show, using complex numbers, that sin(x)+cos(x)=(√2)cos(x-∏/4)

## Homework Equations

cos(x)=(e^(ix)+e^(-ix))/2
sin(x)=(e^(ix)-e^(-ix))/2i

e^ix=cos(x)+isin(x)

## The Attempt at a Solution

I was given the hint that sin(x)=Re(-ie^(ix)), but have thus far not been able to determine its usefullness. I have tried squaring the expression, which (after simplification) yields:

1+sin(2x)

but cannot seem to go further. I assume that the crux of the solution lies in fully expressing sin(x)+cos(x) as purely cos in terms of complex exponentials, but everything I try just brings me back to the original expression. Am I missing a fundamental equivalence between either of these trigonometric functions and a complex number? Any help would be greatly appreciated, thank-you in advance.

(Obviously all work done in trying to solve this problem involves the assumption that I do not know the final answer)

PedroB said:

## Homework Statement

Show, using complex numbers, that sin(x)+cos(x)=(√2)cos(x-∏/4)

## Homework Equations

cos(x)=(e^(ix)+e^(-ix))/2
sin(x)=(e^(ix)-e^(-ix))/2i

e^ix=cos(x)+isin(x)

## The Attempt at a Solution

I was given the hint that sin(x)=Re(-ie^(ix)), but have thus far not been able to determine its usefullness. I have tried squaring the expression, which (after simplification) yields:

1+sin(2x)

but cannot seem to go further. I assume that the crux of the solution lies in fully expressing sin(x)+cos(x) as purely cos in terms of complex exponentials, but everything I try just brings me back to the original expression. Am I missing a fundamental equivalence between either of these trigonometric functions and a complex number? Any help would be greatly appreciated, thank-you in advance.

(Obviously all work done in trying to solve this problem involves the assumption that I do not know the final answer)

Start with the right side of your identity, √2 cos(x - ##\pi/4##). The identity I show just above can be used to rewrite √2 cos(x - ##\pi/4##) in its exponential form. The rest is fairly straightforward but does take a few steps.

## 1. What are trigonometric identities?

Trigonometric identities are equations that relate different trigonometric functions and help to simplify and solve trigonometric expressions.

## 2. How do I prove a trigonometric identity?

To prove a trigonometric identity, you must manipulate one side of the equation using algebraic and trigonometric properties until it is equivalent to the other side.

## 3. How are complex numbers related to trigonometric identities?

Complex numbers can be represented in the form a + bi, where a and b are real numbers and i is the imaginary unit. Trigonometric identities can be used to simplify complex numbers and express them in terms of trigonometric functions.

## 4. Can complex numbers be used to solve trigonometric equations?

Yes, complex numbers can be used to solve trigonometric equations, particularly those involving multiple angles or non-real solutions.

## 5. What are some real-life applications of trigonometric identities and complex numbers?

Trigonometric identities and complex numbers are used in various fields such as engineering, physics, and mathematics to solve problems involving waves, oscillations, and electrical circuits.

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