Trig+Ids+9-27


 * Goals of the Trigonometric Identities Unit:**
 * 1) Be able to find and recreate derivations of "standard" trig identities
 * 2) Use "standard" trig identities to derive new identities
 * 3) Have some of the "standard" identities memorized to help derive others


 * Note: The term "standard" trig identities varies depending on which source you consult.


 * Classes of Trig IDs (Trigonometric Identities):**
 * Reciprocal
 * Quotient
 * Cofunction
 * Pythagorean
 * Negative angle (even/odd)

__Reciprocal Trig IDs__ Memorize these (because most of the time, you are given csc, sec, or cot, and you want sin, cos, or tan; also, there aren't buttons on the calculator for csc, sec, or cot): math $ \frac{1}{sin(x)}=csc(x) $ math math $ \frac{1}{cos(x)}=sec(x) $ math math $ \frac{1}{tan(x)}=cot(x) $ math But you don't have to memorize these because they're pretty obvious: math $ \frac{1}{csc(x)}=sin(x) $ math math $ \frac{1}{sec(x)}=cos(x) $ math math $ \frac{1}{cot(x)}=tan(x) $ math
 * Discussions of Each Class:**

__Quotient Trig IDs__ math $ cot(x)=\frac{cos(x)}{sin(x)} $ math math $ tan(x)=\frac{sin(x)}{cos(x)} $ math Of these, plus the reciprocal ID math $ \frac{1}{cot(x)}=tan(x) $ math you only need to memorize two. Also, the quotient IDs are really definitions rather than identities.

__Cofunction Trig IDs__ math $ sin\left ( \frac{\pi }{2}\left -x\right \right )= cos\left ( x\right ) $ math This says that the sine of the complementary angle equals the cosine of the angle. If you know this one, you can find that math $ cos\left ( \frac{\pi }{2}\left -x\right \right )=sin\left ( x \right ) $ math because math $ cos\left ( \frac{\pi }{2}\left -x\right \right )=sin\left ( \frac{\pi }{2}\left -\left ( \frac{\pi }{2} \right -x\right ) \right ) $ math math $ cos\left ( \frac{\pi }{2}\left -x\right \right )=sin\left ( \frac{\pi }{2}-\frac{\pi }{2} +x\right ) $ math math $ cos\left ( \frac{\pi }{2}\left -x\right \right )=sin(x) $ math This says that the cosine of the complementary angle equals the sine of the angle. Others are: math $ sec\left ( \frac{\pi }{2}\left -x\right \right )=csc(x) $ math math $ csc\left ( \frac{\pi }{2}\left -x\right \right )=sec(x) $ math math $ tan\left ( \frac{\pi }{2}\left -x\right \right )=cot(x) $ math math $ cot\left ( \frac{\pi }{2}\left -x\right \right )=tan(x) $ math You can use graphs to help learn these identities. For example, for math $ sin\left ( \frac{\pi }{2}\left -x\right \right )= cos\left ( x\right ) $ math you can find math $ sin\left ( -\left ( x- \right \frac{\pi }{2}) \right ) $ math by drawing the graph of math $ sin(x) $ math then transforming it into the graph of math $ sin(-x) $ math by flipping it over the x-axis. You can then use the graph of math $ sin\left ( -\left ( x- \right \frac{\pi }{2}) \right ) $ math which flips again over the x-axis and shifts the graph right by math $ \frac{\pi }{2} $ math Then, you can see that it's the same graph as cosine.

__Pythagorean Trig IDs__ The only one you need to memorize is math $ cos^{2}(x)+sin^{2}(x)=1 $ math From there, you can work out the other two, which are found by dividing the given identity by either sine or cosine. math $ \frac{cos^{2}(x)}{sin^{2}(x)}+\frac{sin^{2}(x)}{sin^{2}(x)}=\frac{1}{sin^{2}(x)} $ math math $ cot^{2}(x)+1=csc^{2}(x) $ math

math $ \frac{cos^{2}(x)}{cos^{2}(x)}+\frac{sin^{2}(x)}{cos^{2}(x)}=\frac{1}{cos^{2}(x)} $ math math $ 1+tan^{2}(x)=sec^{2}(x) $ math

__Negative Angle Trig IDs__ math $ sin(-x)=-sin(x) $ math math $ tan(-x)=-tan(x) $ math math $ cos(-x)=cos(x) $ math One way to think about this: anything associated with cosine (so cosine and secant) has the negative dropped. Look at the unit circle: Point (X,Y) has the same cosine as it would if you flipped it over the x-axis, but it would have an opposite tangent and sine, which shows how these identities work.

__Proving Trig IDs__ We got two handouts describing how to prove trig identites. The mains ways are to pick the complicated side and change it, don't move numbers across the equal signs, and multiply by various forms of 1 to make things cancel. Experimentation is important!

Ex: Prove math $ sec(-x)=sec(x) $ math math $ sec(-x)=\frac{1}{cos(-x)}=\frac{1}{cos(x)}=sec(x) $ math Or, for proofs, you can write them out in words. For this proof, you could write: To prove that math $ sec(-x)=sec(x) $ math I picked the first side to change. Sec(-x) is one divided by the cosine of (-x) by the reciprocal identities. Using the graph or the unit circle, that is the same as one divided by the cosine of x, which (by another reciprocal identity) is secant of x.