Pyrhagorean Identities U5 Day 12.5
What is an Identity? Equations are conditional. Ex. 7 = 2x + 3 Only true if… BUT… an Identity is ALWAYS TRUE, regardless of the value of the variable. Ex. 2(x-1) = 2x – 2 True if x is…
Reciprocal Identities sin x = csc x = cos x = sec x = tan x = cot x = You already know these!
Quotient Identities tan θ = So, cot θ = See how many you know already?
Pythagorean Identities sin2x + cos2x = 1 Is this always true? If x is any angle? That’s because… YES! YES!
So… sin2x + cos2x = 1 In the unit circle, x2 + y2 = 1 y = sinq q 1 X = cos q In the unit circle, x2 + y2 = 1 So… sin2x + cos2x = 1
Divide everything by sin2x What about this? Divide everything by sin2x sin2x + cos2 x = 1 ___________________ ________________ ______________ sin2x sin2x sin2x 1 + cot2x = csc2x Another identity… This is the same as… sin2x + cos2x = 1
How about this one? Divide everything by cos2x sin2x + cos2 x = 1 ________________ ________________ ______________ cos2x cos2x cos2x tan2x + 1 = sec2x Another identity… This is the same as… sin2x + cos2x = 1
Pythagorean Identities sin2x + cos2x = 1 1 + tan2x = sec2x 1 + cot2x = csc2x
Use identities to find sin x, cos x, tan x Given sin x = -2/3 and x is in quadrant III, find cos x and tan x. Or…make a triangle
Simplifying Expressions! Ex. 1 sin x cos x – sin x sin x (cos x – 1)
Factoring Trig Expressions Ex.2
Factoring Trig Expressions Ex.3
Simplifying Expressions Ex.4 = csc θ Hint: get a common denominator
Remember the Pythagorean Identity! sin2x + cos2x = 1