FUNDAMENTAL TRIGONOMETRIC IDENTITIES

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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:   $(x+y)^2 = (2-3)^2 = (-1)^2 = 1$
• right side:   $x^2 + 2xy + y^2 = 2^2 + 2(2)(-3) + (-3)^2 = 4 - 12 + 9 = 1$

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: $$\cssId{s22}{(x+y)^2 = (x+y)(x+y) = x^2 + 2xy + y^2}$$ 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 $\,\bigl(\cos t,\sin t\bigr)\,$ 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 at right, 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: $$ \cssId{s56}{\sin^2 t + \cos^2 t = 1^2 = 1} $$ 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: $$ \cssId{s81}{\overbrace{\strut\sin(-t)}^{\text{sine of $-t$}}\ \ \ \overbrace{\strut =}^{\text{is}}\ \ \ \overbrace{\strut -}^{\text{the opposite of}}\ \ \ \overbrace{\strut\sin(t)}^{\text{sine of $t$}}} $$

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Periodic Functions