1. Activity 7: Investigating Compound Angles
Click the picture to continue B A C D

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: x y
Split the angle into radian angles of π/6 + π/4. This is the same as 45o and 30o. The angle is now a compound angle.
We can solve for this right triangle since we have our angle, π/6 and the hypotenuse of 1.
Since we are trying to find the ratio sin(π/6+ π/4), we should identify the sides we need in order to solve this ratio. sin(angle)=OP/HY.
The altitude of the right triangle created with the angle π/4 can be solved since its hypotenuse is cos(π/6).
We are now going to create a right triangle inside the π/6 compound angle where the right angle start on the end of the green arrow.
Given the angles in the smallest right triangle, it is now possible to calculate the altitude.
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 π/4 rads.
We use the cosine ratio:
cos(π/4)=AD/HY
cos(π/4)=Altitude/sin(π/6)
Altitude=sin(π/6)cos(π/4)
Therefore,
sin(75o)=sin(π/6+ π/4)
sin(π/6)cos(π/4)+sin(π/4)cos(π/6)
When we calculate this using special angles:
=0.966
This is the answer we got on the calculator
HYPOTENUSE
π/4
sin(π/6)
sin(π/6)cos(π/4) 1
OPPOSITE
cos(π/6) β
π/6
sin(π/4)cos(π/6)
75o
π/4

5. Activity 7: Investigating Compound Angles
In general, when finding the sine ratio of a compound angle as shown below: sin(A+B) = sinAcosB + sinBcosA
sinAcosB+sinBcosA A B

6. Activity 7: Investigating Compound Angles
Go back to the activity website and complete the rest of the activity online!