7.1 – Basic Trigonometric Identities and Equations. Trigonometric Identities. Quotient Identities. Reciprocal Identities. Pythagorean Identities. sin 2 q + cos 2 q = 1. tan 2 q + 1 = sec 2 q. cot 2 q + 1 = csc 2 q. sin 2 q = 1 - cos 2 q. tan 2 q = sec 2 q - 1.
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Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin2q + cos2q = 1 tan2q + 1 = sec2q cot2q + 1 = csc2q sin2q = 1 - cos2q tan2q = sec2q - 1 cot2q = csc2q - 1 cos2q = 1 - sin2q 5.4.3
Where did our pythagorean identities come from?? Do you remember the Unit Circle? • What is the equation for the unit circle? x2 + y2 = 1 • What does x = ? What does y = ? • (in terms of trig functions) sin2θ + cos2θ = 1 Pythagorean Identity!
Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos2θ sin2θ + cos2θ = 1 . cos2θcos2θ cos2θ tan2θ + 1 = sec2θ Quotient Identity Reciprocal Identity another Pythagorean Identity
Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin2θ sin2θ + cos2θ = 1 . sin2θsin2θ sin2θ 1 + cot2θ = csc2θ Quotient Identity Reciprocal Identity a third Pythagorean Identity
Using the identities you now know, find the trig value. 1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.
3.) sinθ = -1/3, find tanθ 4.) secθ = -7/5,find sinθ
REMEMBER…. TO NUMBER EACH STEP WRITE CLEARLY GO ALL THE WAY TO ONE TRIG VALUE (DON’T LEAVE TAN2X, LEAVE TANX)
Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. b) a) 5.4.5
Simplifing Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x d)
sin x cos x Simplifying trig Identity Example1: simplify tanxcosx tanx cosx tanxcosx = sin x
sec x sec x sin x sinx 1 1 1 = tan x = = x cos x cos x cos x csc x csc x sin x 1 Simplifying trig Identity Example2: simplify
cos2x - sin2x cos2x - sin2x cos2x - sin2x 1 = sec x cos x cos x Simplifying trig Identity Example2: simplify
Example Simplify: Factor out cot x = cot x (csc2 x - 1) Use pythagorean identity = cot x (cot2 x) Simplify = cot3 x
= 1 = sin2 x + cos2x = sin x (sin x) + cos x = sin2 x + (cos x) cos x cos x cos x cos x cos x cos x Example Simplify: Use quotient identity Simplify fraction with LCD Simplify numerator Use pythagorean identity Use reciprocal identity = sec x
Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity
One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this: substitute using each identity simplify
Another way to use identities is to write one function in terms of another function. Let’s see an example of this: This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.
(E) Examples • Prove tan(x) cos(x) = sin(x)
(E) Examples • Prove tan2(x) = sin2(x) cos-2(x)
(E) Examples • Prove
(E) Examples • Prove