Trigonometric Identities
Trigonometric Identities
A trigonometric identity is an equation involving trigonometric functions that holds true for all angles. These identities are commonly called Pythagorean identities, because they come from the Pythagorean theorem.
In general, the Math IIC will test your knowledge of the trigonometric identities by giving you a complex expression that you have to simplify. These questions have more to do with memorizing the identities and being good with algebraic substitution than they do with the theoretical concepts of trigonometry.
For example:
What is ?
To solve a problem like this, use the trigonometric identities to simplify the trigonometric expression into sines and cosines. After you have simplified the expression using the identities, it is quite likely that the expressions will simplify further through the canceling of terms. The simplification of the expression in the example questions proceeds as follows.
Simplify the mess given to you by the problem, and you get sin2.
Here’s another example:
What is tan2 cos2 + 1 – sin2?
Other Trigonometric Identities
About once a test, you may encounter a particularly complicated trigonometric expression. Such expressions can usually be solved using the trigonometric identities we’ve already covered, but the process will be laborious and difficult. However, if you know the special trigonometric identities below, these questions will be much easier and less time-consuming.
We list the special trigonometric identities in the order of usefulness, starting with the ones most likely to be used on the test.
Sum and Difference Formulas
There are three identities for the sum of two angles:
There are also three identities for the difference of two angles:
Try to use these identities in a sample problem:
Simplify the following expression: sin2x + cos2x – sec2x + tan2x.
Using the identities sin2x + cos2x = 1 and 1 + tan2x = sec2x, the expression simplifies in the following way:
Here’s another example to practice with:
If sin a = –cos b = 3/5 and a and b are both in the second quadrant, what is cos (ab)?
If sin a = 3/5 and the angle is in the second quadrant, then cos a = –4/5. Also, if cos b = –3 /5 and b is in the second quadrant, then sin b = 4/5. See the figure below:
Now you can use the subtraction identity to see that = (–4 /5)(– 3/5) + (3/5)( 4/5) = 12/25 + 12/25 = 24/25.
If you are familiar with the formula for cos(ab) and remember the Pythagorean triple 3-4-5, you can save time on a question like this. Otherwise, you have to calculate the arcsine of 3/5 and the arccosine of –3/ 5 in order to find and , respectively. Then you have to subtract from and take the cosine of the resulting angle.
Double Angle Formulas
There are three double angle formulas.
Half Angle Formulas
Of all trigonometric identities, the six half-angle formulas appear most infrequently on the Math IIC.
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