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Study Notes 037:
Trigonometry 11:
By William E. Steinman:
Equations involving trigonometric functions of unknown angles are called identities if they are satisfied by all values of the unknown angle for which the functions are defined.
Huh?
When I read that I feel like the bumblebee that just flew up it’s own butt and turned inside out. I don’t know why simple things look so complicated when we spell them out in a formal statement.
Let’s take the more general case.
In mathematics, an identity is an equation that is satisfied by any number for which the equation is defined.
Let’s look at an example.
The equation x(x -2) = x2 – 2x meets the requirements for an identity.
If we do a simple manipulation the equation comes out to x2 – 2x = x2 – 2x
No matter what value we assign to x the equation will be true.
This means the equation is an identity.
On the other hand, the equation x -2 = 6 is only true for a value of x = 8.
We call this a conditional identity.
This same idea holds for trigonometric identities.
For example:
sin x csc x =1 is an identity.
It is satisfied for every value of x for which the csc x is defined.
sin x = 1 – cos x is not an identity.
It will not be true if x is replaces by the value 300.
Try it.
Sin 300 = 1 – cos 300
0.5 = 1 – 0.866 = 0.134
So, to show that an equation is not an identity only requires that we find one value of the variable for which the equation is not true.
Proving an equation is an identity is a bit more rigorous. This requires that we prove the expressions given in the equation really are equal.
We can approach that in two general ways.
We can substitute known identities and manipulate the equation algebraically to reach an equation that is obviously true.
We have already given an example of that in the above equation x(x -2) = x2 – 2x.
We can also prove that both expressions are equal to a third expression or value.
In this, we must be careful to follow the rules of basic mathematics. We cannot multiply or divide by zero to get a pseudo solution.
In general, to solve a trigonometric equation, be it an identity or a conditional equation, is no different that solving any other equation. We simply solve for the value or values which will satisfy the equation.
That’s enough for now. If you think you would like to get some exercised in solving problems like this, try picking up a copy of Schaum’s Outlines, Trigonometry by Moyer and Ayres. For what you get, Schaum’s Outlines are the best buy out there.
Next time, we can take a look at complex numbers.