Illustrative
Mathematics
F-TF Sum and Difference angle
formulas
Alignments to Content Standards: F-TF.C.9
Task
In this task, you will show how all of the sum and difference angle formulas can be
derived from a single formula when combined with relations you have already learned.
For the following task, assume that the sum angle formula for sine is true. Namely,
sin(θ + ϕ) = sin θ cos ϕ + cos θ sin ϕ.
a. To derive the difference angle formula for sine, write sin(θ − ϕ) as sin(θ + (−ϕ))
and apply the sum angle formula for sine to the angles θ and −ϕ. Use the fact that sine
is an odd function while cosine is even function to simplify your answer. Conclude that
sin(θ − ϕ) = sin(θ) cos(ϕ) − cos(θ) sin(ϕ).
b. To derive the sum angle formula for cosine, use what what you learned in (a) to show
that
cos(θ + ϕ) = cos θ cos ϕ − sin θ sin ϕ.
You may want to start with an exploration of sin( π
2
− (θ + ϕ)).
c. Derive the difference angle formula for cosine,
cos(θ − ϕ) = cos θ cos ϕ + sin θ sin ϕ.
d. Derive the sum angle formula for tangent,
tan(θ + ϕ) =
tan θ + tan ϕ
.
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Illustrative
Mathematics
tan(θ + ϕ) =
tan θ + tan ϕ
.
1 − tan θ tan ϕ
e. Derive the difference angle formula for tangent,
tan(θ − ϕ) =
tan θ − tan ϕ
.
1 + tan θ tan ϕ
IM Commentary
The goal of this task is to have students derive the addition and subtraction formulas
for cosine and tangent, and the subtraction formula for cosine, from the sum formula
for sine. The task provides varying levels of scaffolding, pointing out possible relations
to use early on, but leaving more creative work for the student later. In addition, the
task assumes the sum angle formula for sine and shows how the other sum and
difference formulas must follow.
This text of this problem and its solution assumes familiarity with the Greek letters
theta (θ) and phi (ϕ). However, some teachers or books will use alpha (α) and beta
(β). Still others use Latin letters like u and v or A and B. Instructors should feel free to
change the letters to match those of their source, as the choice of letters is not
important; it is the relationships the letters represent that are useful.
Before embarking on this task, students should be aware that sine is odd (hence
sin(−θ) = − sin(θ)) and cosine is even (hence cos(−θ) = cos(θ)) as found in standard
F-TF.4. Students should know the relations between sine, cosine and tangent found in
standard G-SRT.6. In addition, students should know the relation between
trigonometric values of "complementary" angles found in standard G-SRT.7, (
sin(θ) = cos(π/2 − θ), etc.).
The emphasis of this task is to show how one result can be extended to a family of
results using known relations. This is a central strategy in mathematical thinking, and
illustrates Standards for Mathematical Practices 7 and 8, looking for structure and
making use of repeated reasonin. Along these lines, solutions other than the ones
given here are also viable -- for example, students might prove part (e) from part (d) by
making the "substitution" ϕ → −ϕ (perhaps not in that language).
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Illustrative
Mathematics
Edit this solution
Solution
We assume the sum angle formula for sine is correct:
sin(θ + ϕ) = sin θ cos ϕ + cos θ sin ϕ.
a. To derive the difference angle formula for sine, we write sin(θ − ϕ) as sin(θ + (−ϕ))
to get
sin(θ − ϕ) = sin(θ + (−ϕ))
= sin θ cos(−ϕ) + cos θ sin(−ϕ)
Since cosine is even and sine is odd, we can simplify cos(−ϕ)
sin(−ϕ) = − sin(ϕ). Substituting, we have
= cos(ϕ) and
sin(θ − ϕ) = sin θ cos(−ϕ) + cos θ sin(−ϕ)
= sin θ cos ϕ − cos θ sin ϕ
as desired. That is,
sin(θ − ϕ) = sin(θ) cos(ϕ) − cos(θ) sin(ϕ).
b. To derive the sum angle formula for cosine, use the angle relations
cos(A) = sin( π − A) and sin(A) = cos( π − A) for any angle A. We begin by
2
2
observing that we can write cos(θ + ϕ) as sin(π/2 − (θ + ϕ)). Moreover, we can write
sin(
π
π
− (θ + ϕ)) = sin(( − θ) − ϕ)
2
2
Putting these equalities together we have
cos(θ + ϕ) = sin((
π
− θ) − ϕ)
2
Using the difference angle formula for sine we have
π
cos(θ + ϕ) = sin(( − θ) − ϕ)
2
π
π
= sin( − θ) cos ϕ − cos( − θ) sin ϕ
2
2
= cos θ cos ϕ − sin θ sin ϕ.
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Illustrative
Mathematics
We conclude that
cos θ cos ϕ − sin θ sin ϕ.
cos(θ + ϕ) = cos θ cos ϕ − sin θ sin ϕ.
c. To derive the difference angle formula for cosine, we apply the strategy from (a) and
write
cos(θ − ϕ) = cos(θ + (−ϕ))
= cos θ cos(−ϕ) − sin θ sin(−ϕ).
Using the fact that sine is odd and cosine is even we reduce further to get
cos(θ − ϕ) = cos θ cos(−ϕ) − sin θ sin(−ϕ)
= cos θ cos ϕ + sin θ sin ϕ
We conclude that
cos(θ − ϕ) = cos θ cos ϕ + sin θ sin ϕ.
d. To derive the sum angle formula for tangent, we can write
tan(θ + ϕ) =
sin(θ + ϕ)
cos(θ + ϕ)
Applying the sum angles for sine and cosine, we have
tan(θ + ϕ) =
sin θ cos ϕ + cos θ sin ϕ
cos θ cos ϕ − sin θ sin ϕ
In order get tangent into the right hand side, we can substitute the equations
sin θ = tan θ cos θ and sin ϕ = tan ϕ cos ϕ. With this substitution, we have
sin θ cos ϕ + cos θ sin ϕ
tan θ cos θ cos ϕ + cos θ tan ϕ cos ϕ
=
cos θ cos ϕ − sin θ sin ϕ
cos θ cos ϕ − tan θ cos θ tan ϕ cos ϕ
(cos θ cos ϕ)(tan θ + tan ϕ)
=
(cos θ cos ϕ)(1 − tan θ tan ϕ)
tan θ + tan ϕ
=
1 − tan θ tan ϕ
In the last two lines, we factored the numerator and denominator and the canceled the
common factors. We conclude that
tan(θ + ϕ) =
tan θ + tan ϕ
.
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Illustrative
Mathematics
tan(θ + ϕ) =
tan θ + tan ϕ
.
1 − tan θ tan ϕ
As an alternate route, at the point of
tan(θ + ϕ) =
sin θ cos ϕ + cos θ sin ϕ
cos θ cos ϕ − sin θ sin ϕ
we could divide each term on the right hand side by cos θ cos ϕ to achieve the same
result.
e. To derive the difference angle formula for tangent, we apply the sum angle formula
for tangent to θ + (−ϕ).
tan(θ − ϕ) = tan(θ + (−ϕ))
tan θ + tan(−ϕ)
=
.
1 − tan θ tan(−ϕ)
Since tangent is odd, we can simplify tan(−ϕ)
have
= − tan ϕ. With this substitution, we
tan θ + tan(−ϕ)
1 − tan θ tan(−ϕ)
tan θ − tan ϕ
=
1 + tan θ tan ϕ
tan(θ − ϕ) =
We conclude that
tan(θ − ϕ) =
tan θ − tan ϕ
.
1 + tan θ tan ϕ
F-TF Sum and Difference angle formulas
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