Fundamental Trigonometric Identities

Index card: 58ab

Trigonometry is abundant with identities—they provide important renaming tools when working with trigonometric expressions.

Three of the most basic trigonometric identities are discussed in this section, after a quick review of the word ‘identity’.

What is an ‘Identity’?

An identity is a mathematical sentence that is always true.

Strictly speaking, ‘1 + 1 = 2’ is an identity. However, the word ‘identity’ is typically reserved for a sentence with one or more variables, that is true for every possible choice of variable(s).

For example, $\,(x+y)^2 = x^2 + 2xy + y^2\,$ is an identity from algebra. No matter what real numbers are chosen for $\,x\,$ and $\,y\,,$ the equation is true. For example, letting $\,x = 2\,$ and $\,y = -3\,$ gives:

• Left side: $$\cssId{s17}{(x+y)^2 = (2-3)^2 = (-1)^2 \color{red}{= 1}}$$
• Right side:
  $$ \begin{align} &x^2 + 2xy + y^2 \cr &\quad = 2^2 + 2(2)(-3) + (-3)^2\cr &\quad = 4 - 12 + 9 \color{red}{= 1} \end{align} $$

The left and right sides will always be equal, regardless of the choices made for $\,x\,$ and $\,y\,.$

In this case, it is a simple application of FOIL to see why the equation is always true:

However, identities are not always this obvious—it is not always this easy to prove that a given sentence is an identity!

What Good are Identities?

Identities can provide tremendous renaming power.

For example, the identity above allows the expressions $\,(x+y)^2\,$ and $\,x^2 + 2xy + y^2\,$ to be substituted, one for the other, whenever it is convenient to do so.

In particular, renaming $\,x^2 + 2xy + y^2\,$ as $\,(x+y)^2\,$ (a perfect square) shows that it is always nonnegative. The sum $\,x^2 + 2xy + y^2\,$ doesn't readily reveal that it can't ever be negative—but the perfect square $\,(x+y)^2\,$ does. Different names can reveal different properties of numbers!

The Pythagorean Identity: $\,\sin^2 t + \cos^2 t = 1$

The Pythagorean Identity, $\,\sin^2 t + \cos^2 t = 1\,,$ is perhaps the most used and most famous trigonometric identity. Remember—when a mathematical result is given a special name, there's a reason!

The Pythagorean Identity follows immediately from the unit circle definition of sine and cosine :

• Recall that, in trigonometry, ‘unit circle’ refers to the circle of radius $\,1\,$ that is centered at the origin. The equation of the unit circle is:  $x^2 + y^2 = 1$
• By definition, cosine and sine give the $\,x\,$ and $\,y\,$ values (respectively) of points on the unit circle. That is, for every real number $\,t\,$ (which can be thought of as the radian measure of an angle, if desired), $\,\bigl(\cos t,\sin t\bigr)\,$ is a point on the unit circle.
• Since $\,(\cos t,\sin t)\,$ is on the circle $\,x^2 + y^2 = 1\,,$ it satisfies the equation. That is, substitution of ‘$\,\cos t\,$’ for ‘$\,x\,$’ and ‘$\,\sin t\,$’ for ‘$\,y\,$’ makes the equation true: $$ \cssId{s43}{(\cos t)^2 + (\sin t)^2 = 1} $$

• The expression ‘$\,\sin^2 t\,$’ is a common abbreviation for ‘$\,(\sin t)^2\,$’. (You save writing two parentheses!)

  Similarly, ‘$\,\cos^2 t\,$’ is a common abbreviation for ‘$\,(\cos t)^2\,$’.

  (More generally, this abbreviation is also used for powers $\,3, 4, 5, \ldots\,.$ )

• Using these abbreviations and switching the order of the sum gives the Pythagorean Identity: $$ \cssId{s49}{\sin^2 t + \cos^2 t = 1} $$

Why the Name ‘the Pythagorean Identity’?

Look at the green triangle shown below, in the first quadrant, in the unit circle:

• The bottom leg has length $\,\cos t$
• The other leg has length $\,\sin t$
• The hypotenuse has length $\,1$

A quick application of the Pythagorean Theorem gives:

Voila! The Pythagorean Identity!

Cosine is an Even Function

Recall that even functions have the property that when inputs are opposites, outputs are the same:

• The numbers $\,-t\,$ and $\,t\,$ are opposites, for all real numbers $\,t\,.$
• Their corresponding outputs from a function $\,f\,$ are $\,f(-t)\,$ and $\,f(t)\,.$
• For even functions, these two outputs must (always) be equal:   $\,f(-t) = f(t)\,.$

Cosine has this property:

By definition:

• The terminal point for $\,t\,$ has $x$-value equal to $\,\cos(t)$
• The terminal point for $\,\color{red}{-t}\,$ has $\color{red}{x}$ -value equal to $\,\color{red}{\cos(-t)}$

From symmetry, these two $x$-values are always the same!

Sine is an Odd Function

Recall that odd functions have the property that when inputs are opposites, outputs are also opposites:

• The numbers $\,-t\,$ and $\,t\,$ are opposites, for all real numbers $\,t\,.$
• Their corresponding outputs from a function $\,f\,$ are $\,f(-t)\,$ and $\,f(t)\,.$
• For odd functions, these two outputs must (always) be opposites:   $\,f(-t) = -f(t)\,.$

Sine has this property:

By definition:

• The terminal point for $\,t\,$ has $y$-value equal to $\,\sin(t)$
• The terminal point for $\,\color{red}{-t}\,$ has $\color{red}{y}$ -value equal to $\,\color{red}{\sin(-t)}$

From symmetry, these two $y$-values are always opposites: