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MATH 1113 PreCalculus

Day #27, Trig Identities I

We've already seen that, thanks to the Pythagorean Theorem, the fact that acute angles of a right triangle are complementary, and the circle extensions of the trigonometric functions, it would be perfectly possible to get by with a "table" of trig. values for only one trig relationship (say "sine") for angles between 0o and 90o, and derive all of the other values.  The study of these relationships is the study of trig "identities."  Recall that a mathematical identity is an equation that is true for all values in a particular domain.  For example,

x = sqrt(x2)

is an identity, if the domain is all non-negative real numbers, but it's false if the domain includes the negatives.  The appropriate identity for all real numbers is:

|x| = sqrt(x2), (but this is also not an identity if we allow complex numbers).

Here's a Wikipedia list of many mathematical identities: http://en.wikipedia.org/wiki/List_of_mathematical_identities

In trigonometry, we have a huge catalog of identities, far too many to memorize.  The basic ones can be broken down into categories based on the right triangle or circle relationships that underlie them.  They should not be memorized, but you should be able to easily puzzle them out, based on your familiarity with the basics of trigonometry and geometry.

Right triangle identities:

Just looking at the standard right triangle above, and remembering the basic facts and definitions, we can see the following:

Cofunction identities:

sin(A) = cos(B) = cos(90o- A)  

 cos(A) = sin(B) = sin(90o-A)

sec(A) = csc(B) = csc(90o- A)  

 csc(A) = sec(B) = sec(90o-A)

tan(A) = cot(B) = cot(90o- A)  

 cot(A) = tan(B) = tan(90o-A)

(We've noticed these before.  You just "stand" at the other angle and re-think the SOHCAHTOA bit, recalling that A + B = 90o)

Ratio Identities: (There are a bunch of these, again just based on the definitions)

csc(A) = 1/sin(A)  

sin(A) = 1/csc(A)

sec(A) = 1/cos(A)  

cos(A) = 1/sec(A)

cot(A) = 1/tan(A)  

tan(A) = 1/cot(A)

tan(A) = sin(A)/cos(A)  

cot(A) = cos(A)/sin(A)

(Of course these identities fail whenever you might attempt to divide by zero)

Pythagorean Identities:

Since a2 + b2 = c2 on our right triangle, if we "normalize" by dividing through by the hypotenuse (making the hypotenuse length = 1), we get the most important of the trig identities:

[sin(A)]2 + [cos(A)]2 = 1, or using the common (but confusing) notation:

sin2A + cos2A = 1.

Example:  Try this out, using your calculator, in the case where A = 22o.

sin(22o) = 0.3746         (0.3746)2 = 0.1403
cos(22o) = 0.9272        (0.9272)2 = 0.8597

0.1403 + 0.8597 = 1

People usually remember this as "sine squared plus cosine squared equals 1," but it's also important to remember what it means that you are doing:  Take a number; find the sine of it.  Square the result.  Take the same original number; find the cosine of it.  Square the result.  Then add the two together: you always get "1."

Here's a Geogebra applet demonstrating this:

Applet

By dividing through by "cos2A" you get:

sin2A/cos2A + cos2A/cos2A = 1/cos2A
(sin(A)/cos(A))2 + 1 = (1/cos(A))2

tan2A + 1 = sec2A

In the same way, if you divide the basic identity through by "sin2A" you get:

1 + cot2A = csc2A

So, here are the big 3 "Pythagorean Identities:"

sin2A + cos2A = 1

tan2A + 1 = sec2A

1 + cot2A = csc2A

Next, we'll see the identities that come from the extension of the right triangle definitions to arbitrary real numbers (the "circle identities".)

 

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Page Last Modified: 22 March, 2007

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