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.

Math isn't the end of the world. After some practice, trig identities are second nature.

Reciprocal 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.

Reciprocal identities are easiest to remember as loops. The two ratios that are related will always make the inverse of each other.

Quotient Identities

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.

Quotient identities are expressed as fractions because they are a division problem. Dividing the sine by the cosine of a triangle will always give you the tangent of the triangle.

Pythagorean 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.

It won't take you this long to master Pythagorean identities.

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.

First row: cotangent graph, tangent graph, cosine graph; Second row: secant graph, cosecant graph; Third row: sine graph

Double angle identities

The double angle identities begin the next level of difficulty for the trigonometric identities. Now, rather than worry about one answer for each identity, there may be more than one correct solution for the problem. The double angle identities are as follows:

  • sin(2x) = 2sin(x)cos(x)
  • tan(2x) = 2tan(x)/1-tan^2(x)
  • cos(2x) = cos^2(x)-sin^2(x) OR 2cos^2(x)-1 OR 1-2sin^2(x)

The double angle identity for cos(2x) has three possible answers because it can be reduced by using the Pythagorean identity sin^2(x) + cos^2(x) = 1. By substituting cos^2(x) or sin^2(x) with 1-sin^2(x) or 1-cos^2(x), respectively, the additional two solutions to cos(2x) can be found.

You aren't seeing double. The normal angle you are trying to solve trigonometric ratios with has been doubled, so the identities change, too.

Sum and difference identities

The last set of identities needed in order to pass Pre-Calculus with ease are the sum and difference identities. Personally, I find these to be the most difficult identities to keep track of. But once you see the pattern in the identities, they will not become as big of a threat as they seem. The sum and difference identities are:

  • sin(u+v) = sin(u)cos(v) + sin(v)cos(u)
  • sin(u-v) = sin(u)cos(v) - sin(v)cos(u)
  • cos(u+v) = cos(u)cos(v) - sin(u)sin(v)
  • cos(u-v) = cos(u)cos(v) + sin(u)sin(v)
  • tan(u+v) = tan(u) + tan(v)/1 - tan(u)tan(v)
  • tan(u-v) = tan(u) - tan(v)/1 - tan(u)tan(v)

Now, there are some tricks to help you remember all of that. First, the sine identities. The sign inside the parentheses will match the sign in the drawn out identity. So if it's sin(u+v), the expression it is equal to will also be an addition problem. The cosine identities will have the opposite sign in the expression than what is in the parentheses. So cos(u+v) is a subtraction problem. The tangent identities require some additional thinking. The sign inside the parentheses will match the sign in the numerator of the fraction, but will be the opposite of the one in the denominator. So tan(u+v) will have a plus sign on the top and a minus sign on the bottom. U and v represent any two values.

It seems that the sum and difference identities deal a lot more with the multiplication of sines and cosines than adding or subtracting them.

example problems

Example problem that uses the sum identity of cosine

picture sources

Created By
Kathryn McCoy


Created with images by globochem3x1minus1 - "trihex"

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