Download We want to find a way of handling expressions such as

Compound and Double Angle Identities
Sin(+) and Cos(+) or Sin(-) and Cos(-)
In order to find Sin(A+B), observe the diagram below.
We can use the equation:
So, we need to find the lengths BF and
FE.
Notice that the angle BCA is a right angle. Giving us the equations BC = sin  and
AC = cos .
The angle FBC is also equal to .
The diagram now looks like:
Remember now, that
represents a length. We can use these
two lengths to calculate BF and FE.
In the triangle BCF.
And the triangle ACD.
This means that
Because they are the same length as each other. (FE = CD)
Therefore we can say that
This is the first identity and may be written as:
To find the cosine of a compound angle:
From the diagram we can see that
If we consider the angle BCF:
Therefore:
Because: (FC = ED)
Now consider the triangle ACD:
Combining these two results.
This is another indentity now written as:
You can use the two compound angle formula to find formula for
and
An easy way to do this is to make the following substitutions:
The two formula now become
sin A = sin (A - B)cosB + sinBcos(A – B)
cos A = cos(A – B)cosB – sinBsin(A – B)
We can re-arrange the cos identity to give:
which we can substitute into the sin indentity, to give:
This can now be re-arranged
Multiply by cosB.
Expand the brackets.
Collect terms. Now we
can use Pythagoras'
theorem.
So we get the result
To find the cos formula, substitute this result back into the cos identity.
Expand the brackets.
Re-arrange.
Divide by cosB.
We can now use Pythagoras'
theorem.
So we have the four main formula for compound angles:
You can also find rules for tan, by dividing the above rules.
Divide every term on the top and bottom by
cosAcosB.
Cancel down the fractions.
Write as tan.
So the coumpound tan formula are
Double angle formula are alternative ways of writing sin2x, cos2x, etc. They are useful
for integration and arise from the compound angle formula.
Use the indentities:
These can be adapted to double fomula by replacing the B with another A.
These are the three double angle formulæ:
Notice that there are 3 forms
of the cos formula, related by
the trigonometric
relationship:
The following table contains angles in degrees and radians, as well as there values when
being used with sin, cosinus, and tangent equations.
Θ -degrees
Θ-radians
Sin
Cos
Tan
45°
π/4
√2/2
√2/2
1
60°
π/3
√3/2
½
√3
90°
π/2
1
0
∞
135°
3π/4
√2/2
-√2/2
-1
180°
π
0
-1
0
150°
5π/6
-1/2
-√3/2
√3/2
225°
5π/4
-√2/2
-√2/2
1
30°
π/6
1/2
√3/2
√3/3
Example Questions
These questions are designed to teach you how to solve problems that use the compound
angles, and there Identities.
Part 1 Using exact values
Example: Show that Sin(45 + 90) = Sin(135)
Sin(45 + 90) = Sin(45)*Cos(90) + Cos(45)*Sin(90)
=√2 * 0 + √2 * 1 = √2
= Sin(135)
2
2
2
The following questions will help you further understand this topic. Demonstrate the
following using the exact values as above:
1) Cos(45 + 180))
2) Tan(30 + 90)
3) Sin(180 – 90)
4) Cos(45 - 30)
5) Tan(225 – 90)
Part 2Remember that π can be used in place of actual angle values. For example:
Cos(π- π/2) = Cos(180-90)
Now solve the following questions the same as before, with exact numbers.
6) Sin(3π/4 + 5π/4)
7) Cos(π/4 + π/3)
8) Tan(π + π/2)
9) Sin(5π/4 - 5π/6)
10) Tan(3π/4 - π/3)
Part 3Now apply your knowledge to work in a different direction.
Example: What is the exact value for Sin(75)? Use two exact values you now ie:
Sin(45 + 30) Now you have the same question as you had before.
= Sin(45) * Cos(30) + Cos(45) * Sin(30)
= √2/2 * √3/2 + √2/2 * ½
= √6/4 + √2/4
= √6 +√2
4
Now solve for the exact numbers of the following:
11) 165
12) 105
13) 150
14) 285
15) 315
*** Answers to all the above questions can be found in Appendix A.
For questions or more help refer to the links at the back of this document.
Once these questions have been completed you may continue on and take the
Comprehension test on the page following this one.
Comprehension Test
Compound Angle Identities
1)
2)
3)
4)
5)
6)
7)
Write the six different Compound Angle Identity Equations.
Using Sin find the exact value for 15
Using Cos find the exact value for 105
Using Tan find the exact value for 150
Does Sin(30 + 60) = Sin 90? Prove why or why not.
Does Tan(90 – 30) = Tan 60? Prove why or why not.
a) Solve the following using exact values.
Cos(π/4 + π/3)
b) Solve the same equation using decimals.
c) Compare the two answers, what do you see?
What is the difference between fractional answers and decimals?
8) What is a) Tan (30 + 90)?
b) Sin (45 + 30)?
9) Represent the following in exact values , using algebra.
a) Sin 30 * Tan 180
b) Cos 45 * Tan 60
Bonus!
10) What is Cos(A+B) * Cos(A-B)?
*** Note: The student may be provided with a table of the absolute values, if deemed
necessary by the teacher.
Appendix A:Answers & Solutions
Example Questions
1) Cos (45 + 180) = Cos 45 * Cos 180 – Sin 45 * Sin 180
= √2/2 * -1 - √2/2 * 0
= -√2/2 – 0
= -√2/2
2) Tan (30 + 90) = Tan 30 + Tan 90 .
1 – Tan 30 * Tan 90
= Impossible addition of infinity can’ t be completed then
divided by a multiple of infinity.
3) Sin (180 – 90) = Sin 180 * Cos 90 – Cos 180 * Sin 90
= 0 * 0 - -1 * -1
= 0 - -1
=1
4) Cos (45 – 30) = Cos 45 * Cos 30 + Sin 45 * Sin 30
=√2/2 * √3/2 + √2/2 * ½
= √6/4 + √2/4
= √6 + √2
4
5) Tan (225 – 60) = Tan 225 – Tan 60
1 + Tan 225 * Tan 60
= 1 - √3
1 + 1 * √3
= 1 - √3
1 + √3
6) = √2/2 * -√2/2 + -√2/2 * -√2/2
= -2/4 + 2/4
= 0/4
=0
7) = √2/2 * ½ -√2/2 * √3/2
= √2/4 - √6/4
= √2 - √6
4
8) = Tan π + Tan π/2
1- Tan π * Tan π/2
=0+∞
1- 0 * ∞
=∞
1
=∞
9) = -√2/2 * -√3/2 - -√2/2 * -1/2
= √6/4 -√2/4
=√6 -√2
4
10) = -1 - √3
1 + -1 * √3
= -1 - √3
1 -√3
11 – 15) These questions were designed to give you another way of looking at the same
problem, therefore the same procedure applies. There are too many possible soloutions
to have them all listed.
Comprehension Test
Answers:
1) Sin(A+B) = Sin A * Cos B + Cos A * Sin B
Sin(A-B) = Sin A *Cos B – Cos A * Sin B
Cos(A+B) = Cos A * Cos B – Sin A * Sin B
Cos(A-B) = Cos A * Cos B + Siin A * Sin B
Tan(A+B) = Tan A + Tan B
1- Tan A * Tan B
Tan(A-B) = Tan A – Tan B
1 + Tan A * Tan B
2) √6 - √2
4
3) √2 - √6
4
4) -√3/3
5)
Web page addresses for additional information:
1) http://www.math2.org/math/trig/identities.htm
2) http://www.sosmath.com/trig/Trig5/trig5/trig5.html
3) http://www.easymaths.org/Grade%2012/Trigonometry12/compound_angles.htm