 Identities are statements which are always true, regardless of the angle. They can be used to simplify complicated expressions or formulas. You also need to be able to verify that an identity is true. There are many strategies for doing this: Strategies:
Pythagorean identities are ones based on the Pythagorean theorem. The basic identity is sin2θ + cos2θ = 1 You can see that this works with any angle you choose. For example, on a calculator: sin260° + cos260° = 0.75 + 0.25 = 1 This is called an identity because it will work for any and every angle. When you square the sine of the angle and add that to the square of the cosine of the same angle, you will always get 1. As with all identities, there are many ways to prove that this is always true. In order to prove it, we need to do some algebra without using a specific angle. The definitions of sinθ and cosθ are y/r and x/r, where x2 + y2 = 1 So sin2θ + cos2θ = (y/r)2 + (x/r)2 = y2/r2 + x2/r2 = (y2 + x2)/r2 = (x2 + y2)/r2 = r2/r2 = 1 Identities are used to simplify complicated expressions and formulas in math and the sciences. You can also use them to prove or discover new identities. Here are the three Pythagorean identities:
In Math 30, these are on the formulas sheet you are given. When using them, some people prefer to write out all the rearrangements of each. Here they are:
Let's use these to simplfy some expressions; go on to page two ... |