aim: what are the co-functions and quotient identities in trigonometry?
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Aim: What are the Co-functions and Quotient Identities in Trigonometry?. Do Now:. =. Reciprocal Identities. Quotient Identities. Use in verification of identities and in numerical calculations. Verify the Identity. A trig identity is an equation that is true for all values of variable. - PowerPoint PPT PresentationTRANSCRIPT
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Aim: What are the Co-functions and Quotient Identities in Trigonometry?
Do Now:
cos adjacenthypotenuse
? tan= opposite
adjacent
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Reciprocal Identities
csc 1
sin
sec 1
cos
cot 1
tan
Quotient Identities
1sin
csc
1cos
sec
1tan
cot
sintan
cos
cos
cotsin
Use in verification of identities and in numerical calculations
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Verify the Identity
sin cot cos
1
tasin c
nos
A trig identity is an equation that is true for all values of variable.
verify
csc 1
sin
sec 1
cos
cot 1
tan
sintan
cos
coscot
sin
1si
sin con
cos
s
cos
ssin cos
in
cos cos
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
What does simplifying mean?
Simplify the expression csc tan .
csc tan 1 sin
sin cos
1
cos sec
csc 1
sinReciprocal Identity
sintan
cos
Quotient Identity
Simplifying a trig expressions usually means expressing in terms of sine and cosine.
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Model Problems
csc 1
sin
sec 1
cos
cot 1
tanWrite an expression in terms of sin , cos , or both. Simplify whenever possible.
1. sec • cot
1
cos
1
tan
1
cos
cossin
1
sin
2.
sincos
1sin
tancsc
cos sincos sin
sin2cos
(sin )2 = sin2 usually written
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Model Problems
csc 1
sin
sec 1
cos
cot 1
tan
Find the exact numerical value for each:
1. cot2 30º
1
tan2 300
1
33
2
3
1 tan ENTER
Display: 3
÷ 30
Calculator:
) x2
22cot cotx x
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Model Problems
csc 1
sin
sec 1
cos
cot 1
tan
Find the exact numerical value for each:2. cot 45º • csc 45º
tan 45º = 1
1 • cot 45º • csc 45º
0 2sin45
2
0 1csc45 2
22
cot 45º= 1
2 2
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Trigonometry Co-functions
sin = cos(90º – ) cos = sin(90º – )Co-
sine cosine
tan = cot(90º – ) cot = tan(90º – )Co-
tangent cotangent
sec = csc(90º – ) csc = sec(90º – )Co-
secant cosecant
An trigonometric function of an acuteangle is equal to the co-function of its
complement.
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Model Problems
Write in terms of the co-function the sine of 30o.
sin = cos(90º – ) cos = sin(90º – )Co-
1sin30
2 cos(90 30 ) cos60
If x and y are the measures of two acute angles and tan x = cot y, then
1) x = y + 90 2) x = y – 90
3) x = 90 – y 4) y = x – 90
tan x = cot(90º – x) cot x = tan(90º – x)Co-
y = (90º – x)
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Model Problem
If sin x = cos(x + 20º) and x and (x + 20º)are the measure of two acute angles, find x.
sine and cosine are co-functions
If the sine of an acute angleis equal to the cosine of another
acute angle, the angles mustbe complementary.
x + 20º + x = 90º2x + 20º = 90º
2x = 70ºx = 35º x + 20º = 55º
sin 35º = .57357… cos 55º = .57357… √
Aim: Co-functions & Quotient Identities Course: Alg. 2 & Trig.
Model Problems
Which expression is NOT equal to the other three expressions?
2 cot sin 2cosA. B. C. D.
1 1tan sincos2 2