Activity 7: Investigating Compound Angles

1
Activity 7 Investigating Compound Angles
Click the picture to continue
2
Activity 7 Investigating Compound Angles

• In the introduction of the activity, we conclude
  that the best way to find the exact value of
  sin75o was to write is as a sum of two angles

• We know that

• So, if then,

3
Activity 7 Investigating Compound Angles

• When we try to get an approximate value on the
  calculator we get

• Why do we get two different answers?

• We have to graph this angle in standard position
  to verify which answer is correct.

4
Activity 7 Investigating Compound Angles

• Let us place this angle in standard form where
  radius is 1

We can solve for this right triangle since we
have our angle, p/6 and the hypotenuse of
1.
Given the angles in the smallest right triangle,
it is now possible to calculate the
altitude.
We use the cosine ratio cos(p/4)AD/HY cos(p/4)A
ltitude/sin(p/6) Altitudesin(p/6)cos(p/4)
Therefore, sin(75o)sin(p/6 p/4) sin(p/6)cos(p/4)
sin(p/4)cos(p/6) When we calculate this using
special angles 0.966 This is the answer we
got on the calculator
Split the angle into radian angles of p/6 p/4.
This is the same as 45o and 30o. The angle is
now a compound angle.
Since we are trying to find the ratio sin(p/6
p/4), we should identify the sides we need in
order to solve this ratio. sin(angle)OP/HY.

We are now going to create a right triangle
inside the p/6 compound angle where the right
angle start on the end of the green arrow.
The altitude of the right triangle created with
the angle p/4 can be solved since its hypotenuse
is cos(p/6).
To solve for the second altitude we must find the
angle denoted by the blue dot
Using simple geometry and creating a pair of
parallel lines we see that the blue dot and the
red dot must sum up to 90o. Based on alternate
angles, the blue dot must be equal to p/4
rads.
HYPOTENUSE
p/4
sin(p/6)
sin(p/6)cos(p/4)
OPPOSITE
1
cos(p/6)
p/6
ß
sin(p/4)cos(p/6)
75o
p/4
5
Activity 7 Investigating Compound Angles

• In general, when finding the sine ratio of a
  compound angle as shown belowsin(AB)
  sinAcosB sinBcosA

sinAcosBsinBcosA
A
B
6
Activity 7 Investigating Compound Angles

• Go back to the activity website and complete the
  rest of the activity online!