What are trigonometric identities?
Trigonometric identities are not a form of torture, despite arguments from Pre-Calculus that would suggest otherwise. Identities are simple statements that have solutions true for all real numbers; trigonometric identities deal with sine, cosine, tangent, and their inverses and are used to solve trigonometric proofs. Trigonometric proofs work exactly like geometric or algebraic proofs. A trigonometric proof could be proving two sides of an equation are equal, or finding solutions to an equation. All this can be done by memorizing the identities.
Reciprocal identities prove that a trigonometric ratio has an inverse relationship to other trig functions. These are:
- cot(x) =1/tan(x)
- tan(x) = 1/cot(x)
- sin(x) = 1/csc(x)
- csc(x) = 1/sin(x)
- cos(x) = 1/sec(x)
- sec(x) = 1/cos(x)
As you can see, reciprocal identities tend to become redundant and obvious. However, it is still helpful to review these identities as they become essential to problem solving later on.
Like the reciprocal identities, the quotient identities can also be found in a basic level trig course. The quotient identities are the ratios used in order to find the tangent and cotangent of a right triangle. The ratios are:
- tan(x) = sin(x)/cos(x)
- cot(x) = cos(x)/sin(x)
Reciprocal identities and quotient identities are easy enough to understand. The real struggle comes in when you have to memorize the more abstract identities.
Pythagorean identities are some of the easiest identities to use. They relate back to the right triangle and the Pythagorean theorem. The identities are:
- sin^2(x) + cos^2(x) = 1
- tan^2(x) + 1 = sec^2(x)
- 1 + cot^2(x) = csc^2(x)
These identities are most helpful when the equation becomes extremely long and things can be simplified to make it easier to understand.
Even and odd identities
These next identities are more easily understood when you think of the graphs of these functions. When a function is said to be even, both ends of the graphed line will continue on in the same direction. When one goes up, the other goes up, too. For an odd function, the ends will point in opposite directions. When one end goes up, the other goes down. Knowing this, we can get a better understanding of why the following identities are true. The even and odd identities are:
- sin(-x) = -sin(x)
- csc(-x) = -csc(x)
- tan(-x) = -tan(x)
- cot(-x) = -cot(x)
- cos(-x) = cos(x)
- sec(-x) = sec(x)
The even functions are the cosine and secant functions. Their identity states that when x is a negative number, the answer will be a positive value. Sine, cosecant, tangent, and cotangent are all odd values. When x is negative, the answer will also be negative.