Objectives:
1. To use the CTMethod to solve Q3
on the TRIG*STAR
2. To use sum and
difference formulas
Assignment:
• Practice TRIG*STAR
problems
You will be able to use the
CT-Method to solve Q3 on
the TRIG*STAR
3. On a quadrilateral, you’re given two consecutive sides and
the opposite angle. You have to find the other two sides and
the diagonal.
1. Store m<BAD as A.
y
x
A
A–x
2. Label <BAC as x
and <CAD as A – x.
3. Label diagonal AC
as y.
4. Set up a system of
equations to find x
and y.
3. On a quadrilateral, you’re given two consecutive sides and
the opposite angle. You have to find the other two sides and
the diagonal.
sin x  504.27
y
sin  A  x   265.56
y
y
x
A
(Solve for y)
504.27
sin x
y
y  sin265.56
 A x 
A–x
3. On a quadrilateral, you’re given two consecutive sides and
the opposite angle. You have to find the other two sides and
the diagonal.
6. Graph system of
equations to find
point of
intersection.
y
x
A
y  504.27
sin x
y  sin265.56
 A x 
A–x
3. On a quadrilateral, you’re given two consecutive sides and
the opposite angle. You have to find the other two sides and
the diagonal.
6. Graph system of
equations to find
point of
intersection.
y
x
A
A–x
3. On a quadrilateral, you’re given two consecutive sides and
the opposite angle. You have to find the other two sides and
the diagonal.
6. Graph system of
equations to find
point of
intersection.
y
x
A
A–x
3. On a quadrilateral, you’re given two consecutive sides and
the opposite angle. You have to find the other two sides and
the diagonal.
7. Record the value of
y as AC. The values
of x and y are
stored as X and Y.
y
x
A
A–x
3. On a quadrilateral, you’re given two consecutive sides and
the opposite angle. You have to find the other two sides and
the diagonal.
8. Solve for AB using
P-Thag or
SohCahToa.
y
x
A
A–x
On the third question, label the diagram to match the one below.
Then solve a system of equations of this form:
y
y
x
A
A–x
Top
sin x
Right Side
y
sin  A  x 
Solve using the CT-Method.
You will be able to use
sum and difference
formulas
sine
cosine
tangent
sin(u  v)  sin u cos v  cos u sin v
sin(u  v)  sin u cos v  cos u sin v
cos(u  v)  cos u cos v  sin u sin v
cos(u  v)  cos u cos v  sin u sin v
tan u  tan v
tan(u  v) 
1  tan u tan v
tan u  tan v
tan(u  v) 
1  tan u tan v
Write 15° as the sum or difference of two angles
from the original unit circle.
Use a sum or difference formula to find the
exact value of sin(15°).
Solve the system of equations by using a difference formula.
y  504.27
sin x
y  sin265.56
 A x 
Let’s prove cos (u – v) = cos u cos v + sin u sin v
First we’ll set up a diagram:
• On a unit circle, construct
angles u and v such that
u>v
• Now construct the angle
u–v
• Assign variables for each
point
Let’s prove cos (u – v) = cos u cos v + sin u sin v
Now draw in two segments,
AC and BD.
• Notice these segments
create 2 isosceles
triangles: ΔOAC and ΔOBD
• Since <AOC is congruent to
<DOB, the two triangles
are congruent
• Thus, AC = BD
Let’s prove cos (u – v) = cos u cos v + sin u sin v
AC  BD
(By the distance formula)
 x2 1   y2  0
2
2

 x3  x1    y3  y1 
2
(By squaring both sides)
 x2  1   y2  0 
2
2
  x3  x1    y3  y1 
2
2
2
Let’s prove cos (u – v) = cos u cos v + sin u sin v
AC  BD
(By the distance formula)
 x2 1   y2  0
2
2

 x3  x1    y3  y1 
2
2
(By squaring both sides)
 x2  1   y2  0 
2
2
  x3  x1    y3  y1 
2
2
(By simplifying both sides)
x2 2  2 x2  1  y2 2  x32  2 x3 x1  x12  y32  2 y3 y1  y12
Let’s prove cos (u – v) = cos u cos v + sin u sin v
(Since point B, C, and D lie on the unit circle)
x2 2  y2 2  1
x32  y32  1
x12  y12  1
x2 2  2 x2  1  y2 2  x32  2 x3 x1  x12  y32  2 y3 y1  y12
(By Substitution)
2 x2  1  1  2 x3 x1  1  2 y3 y1  1
(By simplifying)
2 x2  2 x3 x1  2 y3 y1
Let’s prove cos (u – v) = cos u cos v + sin u sin v
2 x2  2 x3 x1  2 y3 y1
(By dividing both sides by −2)
(By unit circle definitions)
x2  x3 x1  y3 y1
cos  u  v   x2
cos u  x3
cos v  x1
sin u  y3
sin v  y1
(By Substitution)
cos  u  v   cos u cos v  sin u sin v
After proving cos (u – v) = cos u cos v + sin u sin v :
Prove cos (u + v):
1. Let u + v =
u – (−v)
2. Use substitution
and even/odd
identities:
Prove sin (u ± v):
1. Let u = π/2 – x
in cos (u ± v)
2. Use cofunction
identities to turn
cosine into sine:
cos   x   cos x
cos  2  x   sin x
sin   x    sin x
sin  2  x   cos x
Prove tan (u ± v):
1. Use quotient
identity:
tan x 
2.
sin x
cos x
Use substitution
with the sum
and difference
formulas for
sine and cosine
Objectives:
1. To use the CTMethod to solve Q3
on the TRIG*STAR
2. To understand
Assignment:
• Practice TRIG*STAR
problems