Sec 4.3a Trig Identities
Trig Identities
Identity: A true statement.
Trig Identity: A true statement about the
relationship between trig functions.
Relationships we have already seen:
Quotient Identities
tanθ = sinθ
cosθ
cotθ = cosθ
sinθ
Reciprocal Identities
sinθ = 1/cscθ
cscθ = 1/sinθ
cosθ = 1/secθ
secθ = 1/cosθ
tanθ =1/cotθ
cotθ =1/tanθ
Sec 4.3a Trig Identities
New Identities
sinA = a/c
cosB = a/c
B
c
a
C
b
A
so, sinA = cosB
A + B = 90∘
A and B are complimentary angles
sinθ= cos(90∘ -θ)
This is also true for tan and cot:
tanA = a/b = cotB
tanθ = cot(90∘ -θ)
Cofunction identities
sin(90∘ -θ)= cosθ
tan(90∘ -θ) = cotθ
cos(90∘ -θ)= sinθ
cot(90∘ -θ) = tanθ
sec(90∘ -θ) = cscθ
csc(90∘ -θ) = secθ
In Radians:
sin( π/2-θ)= cosθ
cos(π/2 -θ)= sinθ
tan(π/2 -θ) = cotθ
cot(π/2 -θ) = tanθ
sec(π/2 -θ) = cscθ
csc(π/2 -θ) = secθ
Sec 4.3a Trig Identities
Pythagorean Identities
hyp
opp
θ
adj
opp2 + adj2 = hyp2
If you divide both sides by hyp2:
opp2 + adj2 = hyp2
hyp2 hyp2 hyp2
or, more commonly,
sin2θ + cos2θ = 1
We can get other trig identities from this one.
sin2θ + cos2θ
sin2θ sin2θ
=1
sin2θ
1 + cot2θ = csc2θ
and similarly,
sin2θ + cos2θ = 1
cos2θ cos2θ cos2θ
tan2θ + 1 = sec2θ
Sec 4.3a Trig Identities
Use cofunction identities to evaluate the
expression without a calculator:
1.
cos223
+ cos267
2.
cos210 - sin215 + cos280 -sin275
Solve for x:
3.
sin2 (π/3) + sin2 x = 1
4.
sin2 20 + sin2 x = 1
Sec 4.3a Trig Identities
Examples:
Use trig identities to solve the following. Assume
all angles are acute angles in a right triangle:
1. cotθ = 2 find:
a) tanθ
b) sinθ
c) secθ
d) tan(90-θ)
2. Use trig identities to transform the left side
of the equation into the right side.
2
2
sec θ - tan θ = 1
2
2
2
cot θ(1-cos θ) = cos θ
tanθ + cotθ = cscθsecθ