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identities to know, how to use them, verifying identities, solving trig equations, determining algebraic expressions, and key notes

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GOUfWufGFJNiiAcVAknE

Study note

Mira

United States of America

DEFAULT (US)

DEFAULT

Pre-Calculus

Trigonometry

trigonometric identities

Trigonometric identities on the unit circle

unit two STUDY guide
IDENTITY BASICS
x+1=X or
• true for any input variable →
tane
4
•rationalizing is also an identity: no variables but still always works
the same graph! (infinitely many solutions)
also will have
TRIG FUNCTION IDENTITIES
sine
quotient identities: cose = tane;
cose/ sine = cote
reciprocal identities: sine = "/csco, csce = //sine; case = "/sece, sece = /cose;
tane= //cote, cote = /tane
0
O
pythagorean identities: sin² 0 + cos² @ = 1; sec²0 -tan ²³ 0 = 1; csc ²0 - cot ²0 = 1
even/odd identities: sin or tan or csc or cot of -0 = - trige; cos or sec-0 = trige
• cofunction identitles: sin (7/2₂-0) = cos ; cos (1/₂2 - 0) = sine ; tan (1/₂2 - 0) = cote
cot (2-0)
= tane; csc (π/₂ - 0) = sece; sec (1/₂-0) = CSCO
•sum/difference identities: sin (A+B) = sin A Gas B ± sinB cos A; cos (A ±B) = COSA COSB I
sin Asine; tan (A + B) = tan at tanB/1 = tan A tank
double angle identities: sin (20) = 2sing cose; cos (20) = cos ²0 - sin ²0;
cos (20) = 2 cos² -1; cos(20) = 1- sin²0; tan (20) = stane / 1-tan²0
• nalf angle identities: sin(012) = +7√1- cos 6 ; cos (0/2) = +7
1+cose
2
2
VERIFYING TRIG IDENTITIES
1. simplification of expressions:
↳ use substitution with identities to allow cancelling or factoring
cot ²x
tanx tan²x
sine
cose
ex: cotx
tanx
=
tanx
4
L
tanx
=
↳also allows us to solve for individual trig without drawing a triangle
ex: if tanx = 2, find cotx → tanx = //cotx → ½/₂
2. verifying with equations:
start with complicated side, ONLY make changes to ONE SIDE
combine/separate fractions, Factor, or substitute identities until both
sides look the exact same
YOU CANNOT square both sides, multiply by common denominator,
ada/ subtract on both sides → If It gets too complicated, start over!
SOLVING TRIG EQUATIONS
• not identities! only hald true for certain values of x
either told to solve on an interval, or general solutions
giving solutions:
OR XE {.........} unit two study
to study guide
SOLVING TRIG EQUATIONS (continued)
• we're allowed to manipulate both sides of the equation
4 but we are NOT allowed to multiply / divide by any variable
• always end by applying inverse trig_relation
to both sides
↳ relation because for any ? all angles, not just in restricted domain
solving for general solutions
4 find answers on interval of one period (either 2πt or it)
add (period) K to each angle right after taking inverse
on interval x € (0, 2TT)
J
☆ ex: sin (3x) = -1 → solve
1-sin-¹ (-1) = 3x
31
2.2
= 3x
-
or find coterminal up to 6TC (because 3x 210 = 6πt)
3TV/2 + 415/2 = 150/2 1/3 = 710/6
-
add period: 3. ³1+ 2KCK =
31. 12 = 300-10
4. =/32 (³1 2 + 2πCK) = x* then solve up to 6r.
7 / 2 + 4Th / 2 = 11T / 2 · 1/3 =
- 1115/65
DETERMINE ALGEBRAIC EXPRESSIONS
composite trig functions → araw right triangle + label with known info
↳ express inner term as a single variable
4 label on triangle
us solve for missing side
A₂
2
don't forget to include original solution!
PC/2, 710/6, 1175/6
X=
Us replace outer trig function with values from triangle
ex: cos C2csc-(₂)]
1- CSC-¹ (¹/₂) = 0 →
cos (20)
2. Cos (20) = 1- 2 sin ²0
↳ 1-2 (²x) ² + 1-¾/x² + x²-8 = cos [2Csc.¹ (₂)]
• CSC = hyp/opp → x= hyp, 2= opp
aaj= √√x²-4
0
KEY NOTES
• always express a trig equation using trig functions with common argument
• the easiest equations to solve involve only one type of trig function
☆ if you square both sides or multiply by a factor, you MUST check
for extraneous solutions!
including a variable
o when taking the inverse of a trig function, find ALL solutions
before moving
check whether the
to the next step
equation can be
factored as a
Polynomial

trigonometric identities

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trigonometric identities

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

13 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying