1
TRIGONOMETRY IDENTITIES
From the equation of the unit circle:
1
x2 + y2 = 1
y
Therefore (in terms of θ):
θ
1
x
 cos2 
 sin 2 
From the diagram, locate each angle and simplify each expression (in terms of sin and/or cos):
sin(π/2–θ) = __________; sin(π–θ) = __________; sin(–θ) = _________
cos(π/2–θ) = _________; cos(π–θ) = __________; cos(–θ) = _________
In terms of θ, area of the triangle ____________________
PROVE IT.
a
h
θ
b
A
Focus on Angle A with angles α, β, and α+β,
to prove the sin(α+β) identity.
START:
Area of triangle ABC = Area ABH + Area ACH
β α
1
C
H
sin(α + β) = ___________________________________________
sin(α –β) = sin(α + –β) = __________________________________
___________________________________
B
cos(α + β) =
_____________________________
___________________________________
___________________________________
cos(α –β) = cos(α + –β) = __________________________________
___________________________________
tan(α + β) = __________________________
= ________________________________________
tan(α – β) = __________________________
= ________________________________________
sin(2α) = sin(α + α) = _______________________________
______________________________
cos(2α) = cos(α + α) = _______________________________
OR _______________________ OR ____________________
tan(2α) =
_______________________________
______________________________
E
BONUS – On this Unit Circle, determine each function as a distance.
F
Sin θ = _______
Cos θ = _______
C
1
θ
0
D
Tan θ = ______
Cot θ = _______
Sec θ = ______
Csc θ = ______
y
x
A
B
TEAM ROUND – TRIG IDENTITIES
1.
In terms of sin and cos, simplify: sin(x-y)cos y + cos(x-y) sin y
2.
If sin(x) = 3 cos (x), than what is sin(x) cos(x) ?
3.
ABCD is a square and M and N are midpoints of BC and CD respectively.
What is sin MAN ?
4.
If sin 2x sin 3x = cos 2x cos 3x, then what is the least positive value of x in degrees?
5.
Evaluate: log10(tan 1o) + log10(tan 2o) + log10(tan 3o) +…+ log10(tan 88o) + log10(tan 89o).
6.
If tan α and tan β are the roots of x2 – px + q=0 and cot α and cot β are the roots of
x2 – rx + s = 0, then express rs in terms of p and q.
7.
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is 16, and one
of the base angles is arcsin(0.8). Find the area of the trapezoid.
8.
If tan x + tan y = 25 and cot x + cot y = 30, what is tan(x + y)?
9.
In triangle ABC, tan(CAB) = 22/7 and the altitude from A divides BC into segments of length 3 and 17.
What is the area of ABC?
10.
In tetrahedron ABCD, edge AB has length 3. The area of face ABC is 15 and the area of face ABD is 12.
These two faces meet each other at a 30o degree angle. Find the volume of the tetrahedron.
WORKSHEET ANSWERS
Therefore (in terms of θ):
sin2θ + cos2θ = 1
 cos2 
 sin 2 
1 + cot2θ = csc2θ
tan2θ + 1 = sec2θ
From the diagram, locate each angle and simplify each expression (in terms of sin and/or cos):
sin(π/2–θ) = cos(θ) ; sin(π–θ) = sin(θ) ;
sin(–θ) = -sin(θ)
cos(π/2–θ) = sin(θ) ; cos(π–θ) = -cos(θ) ; cos(–θ) = cos(θ)
In terms of θ, area of the triangle = ½ ab sin(θ)
PROVE IT.
Area = ½ hb , but h/a = sin(θ) , so Area = ½ ab sin(θ)
a
h
θ
b
Focus on Angle A with angles α, β, and α+β,
to prove the sin(α+β) identity.
START:
Area of triangle ABC = Area ABH + Area ACH
½ AC*AB*sin(α+β)
sin(α+β)
= ½ AC*AH*sin(β) + ½ AB*AH*sin(α) C
=
sin(α + β) = sin α cos β + sin β cos α
sin(α –β) = sin(α + –β) = sin α cos -β + sin -β cos α
= sin α cos β - sin β cos α
A
β α
1
H
B
cos(α + β) = sin(90 - (α + β)) = sin(( 90 – α) - β))
= sin ( 90 – α) cos β - sin β cos ( 90 – α) = cos α cos β - sin β sin α
cos(α –β) = cos(α + –β) =cos α cos -β - sin α sin -β =
cos α cos β + sin α sin β
tan(α + β) =
–
–
tan(α – β) = tan(α + – β)
–
sin(2α) = sin(α + α) =
=
cos(2α) = cos(α + α) = cos a cos a - sin a sin α = cos2 a – sin2 a
E
OR
2 cos2 a – 1
tan(2α) =
tan( a + a) =
1 – 2 sin2 a
–
BONUS – On this Unit Circle, determine each function as a distance.
F
Sin θ = AC
Cos θ = OA
C
1
θ
0
OR
D
Tan θ = BD
Cot θ = EF
Sec θ = OD
Csc θ = OF
y
x
A
B
TEAM ROUND – TRIG IDENTITIES - ANSWERS
1.
AMC83 #11 In terms of sin and cos, simplify: sin(x-y)cos y + cos(x-y) sin y
sin(x-y)cos y + cos(x-y) sin y = [sin(x) cos(y) – sin(y) cos(x)] cos(y) + [cos(x) cos(y) + sin(x) sin(y)] sin(y)
= sin(x) cos2(y) + sin(x) sin2(y) = sin(x)
2.
AMC88 #13 If sin(x) = 3 cos (x), than what is sin(x) cos(x) ?
If sin x = 3 cos x, tan(x) = 3. Then, sin(x) = 3/√
3.
4.
. sin(x) cos(x) = 3/10
AMC87 #14 ABCD is a square and M and N are midpoints of BC and CD respectively.
What is sin MAN ?
Area of triangle AMN = ½ * √
OR
and cos(x) = 1/√
√
= 4 – 1 – 1 – ½ ; sin(MAN) = 3/2 * 2/5 = 3/5
Let ∠NAD = ∠MAB = α , AB=2, and BM=1.
sin(MAN) = sin(π/2 - 2 α ) = cos(2 α) = 2 cos2( α ) – 1 = 2(2/√
– 1 = 3/5
AMC84 #15 If sin(2x) sin(3x) = cos(2x)cos(3x), then what is the least positive value of x in degrees?
cos(2x)cos(3x) - sin(2x) sin(3x) = 0 or cos(5x) = 0, x = 90/5 = 18
5.
AMC87 #20 Evaluate: log10(tan 1o) + log10(tan 2o) + log10(tan 3o) +…+ log10(tan 88o) + log10(tan 89o).
= log10(tan 1o) (tan 89o) + log10(tan 2o)(tan 88o) +….+log10(tan 45o)
= log10(tan 1o) log10(cot 1o) + log10(tan 2o) (cot 2o) +….+log10(tan 45o)
= log10 1 + log10 1 + …+log10 1 = 0
6.
AMC83 #20 If tan α and tan β are the roots of x2 – px + q=0 and cot α and cot β are the roots of
x2 – rx + s = 0, then express rs in terms of p and q.
By the theorems of sums and products of roots of polynomials: p = tan α + tan β ; r = cot α + cot β ;
q = tan α * tan β ; and s = cot α * cot β
rs = (cot α + cot β ) cot α * cot β =
=
cot α * cot β
*1
= p/
7.
AMC88 #24 An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is
16, and one of the base angles is arcsin(0.8). Find the area of the trapezoid.
Let length of shorter base = y and length of each leg = x. Since sin(α) = 0.8, the legs of the right triangle
with hypotenuse x are 0.6x and 0.8x. Since the sides of the trapezoid are tangent to the circle, the sums of the
length of opposite sides are equal. Thus, 2y + 1.2x = 2x and y + 1.2x = 16. Solving, y=4 and x=10. Area = 80
8.
AIME86 #3 If tan x + tan y = 25 and cot x + cot y = 30, what is tan(x + y)?
Note: cot x + cot y =
Thus, 30 =
or
tan(x + y) =
9.
AIME88 #7 In triangle ABC, tan(CAB) = 22/7 and the altitude from A divides BC into segments of
length 3 and 17. What is the area of ABC?
Let P be the intersection of BC and the altitude from A, let ∠CAP = α and ∠BAP = β.
tan(α) = 3/h and tan(β) = 17/h ; tan(CAB) = 22/7 =
( )( )
Solving: 11h2 – 70h – 11*51 = 0 = (11x + 51)(x – 11). x = 11. Area = 110
10.
AIME84 #9 In tetrahedron ABCD, edge AB has length 3. The area of face ABC is 15 and the area of
face ABD is 12. These two faces meet each other at a 30o degree angle. Find the volume of the tetrahedron.
Let V = volume and h = length of altitude from D. V = 1/3 * h * Area(ABC) = 5h.
Let H be foot of altitude from D. Select K on AB so that DK⏊AB and HK⏊AB.
Area(ABD) = ½ * AB*DK = 12. DK = 24/3 = 8. Since DHK is a 30-60-90 triangle, h = 4. V = 20