Sum and difference formulae for sine and cosine
Consider angles α and β with α > β. These angles identify points on the
unit circle, P (cos α, sin α) and Q(cos β, sin β).
Elementary Functions
Part 5, Trigonometry
Lecture 5.1a, Sum and Difference Formulas
Dr. Ken W. Smith
Sam Houston State University
2013
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Sum and difference formulae for sine and cosine
Sum and difference formulae for sine and cosine
The distance between P and Q is, by the Pythagorean theorem, the
square root of
We can expand this out, algebraically, and get
d(P Q)2
= (cos α −
cos β)2
+ (sin α −
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d(P Q)2 = (cos2 α−2 cos α cos β +cos2 β)+(sin2 α−2 sin α sin β +sin2 β).
sin β)2 .
which we can rewrite (using the Pythagorean identity) as
d(P Q)2 = 2 − 2 cos α cos β − 2 sin α sin β
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Sum and difference formulae for sine and cosine
Sum and difference formulae for sine and cosine
Let’s rotate this picture clockwise through the angle −β so that the point
Q becomes Q0 (1, 0) lying on the x-axis. The point P rotates into the
point P 0 (cos(α − β), sin(α − β))
The distance from P 0 to Q0 is the same as the distance from P to Q.
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Sum and difference formulae for sine and cosine
Sum and difference formulae for sine and cosine
The distance from P 0 to Q0 is the square root of
Since the line segments P Q and P 0 Q0 are congruent, then we know that
d(P Q)2 = d(P 0 Q0 )2 .
d(P 0 Q0 )2 =
(1−cos(α−β))2 +(sin(α−β))2 = 1−2 cos(α−β)+cos2 (α−β)+sin2 (α−β))
= 2 − 2 cos(α − β).
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Sum and difference formulae for sine and cosine
Sum and difference formulae for sine and cosine
D2 = 2 − 2 cos α cos β − 2 sin α sin β
In mathematics, if we arrive at the same value through two different
computations, we always have valuable information. Here we can equate
d(P Q)2 and d(P 0 Q0 )2 and simplify.
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D2 = 2 − 2 cos(α − β)2
Sum and difference formulae for sine and cosine
Sum and difference formulae for sine and cosine
So
We create a more memorable form of this equation if we replace β by −β
and use the fact that sine is an odd function while cosine is an even
function.
2 − 2 cos α cos β − 2 sin α sin β = 2 − 2 cos(α − β)2 .
Write cos(α + β) = cos(α − (−β)) and replace β by −β in our equation
for cos(α − β) to see that
We may divide both sides by 2 and solve for cos(α − β).
cos(α − β) = cos α cos β + sin α sin β
cos(α + β) = cos α cos(−β) + sin α sin(−β) = cos α cos(β) − sin α sin(β)
This is an important result.
This is an equation we want to record and use on a regular basis.
cos(α + β) = cos α cos β − sin α sin β
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(1)
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Sum and difference formulae for sine and cosine
Sum and difference formulae for sine and cosine
π
− θ) (since sine and cosine are
2
complementary functions!) then we can write
If we note that sin(θ) = cos(
Therefore we have a sum-of-angles equation for the sine function:
π
π
sin(α + β) = cos( − (α + β)) = cos(( − α) − β).
2
2
sin(α + β) = sin α cos β + cos α sin β
π
Using the sum-of-angles formula above, with regards to the angles − α
2
and β, we have
π
sin(α + β) = cos(( − α) − β)
2
= cos(
(2)
If we need, we may replace β by −β to create a difference equation:
sin(α − β) = sin α cos β − cos α sin β
π
π
− α) cos β + sin( − α) sin β = sin α cos β + cos α sin β.
2
2
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Sum and difference formulae for sine and cosine
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Sum and difference formulae for sine and cosine
I don’t memorize these identities.
Here they are again:
(That’s one reason I’ve been attracted to mathematics – if one
understands the math, one doesn’t need to memorize!)
cos(α + β) = cos α cos β − sin α sin β
In the undergraduate classes that I teach at Sam Houston State University,
I will provide students with the sum of angle formulas when they are
needed.
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sin(α + β) = sin α cos β + cos α sin β
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Sum and difference formulae for sine and cosine
Sum and Difference Formulas
Here they are as difference of angles
In the next presentation, we will look at some applications of these sum
and difference formulas.
cos(α−β) = cos α cos β+ sin α sin β
(End)
sin(α−β) = sin α cos β− cos α sin β
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A reduction formula
Let P (cos θ, sin θ) be a point on the terminal side of angle θ.
←→
Any point (a, b) on the line OP satisfies the equations
cos θ =
Elementary Functions
√ a
,
a2 +b2
sin θ =
√ b
.
a2 +b2
Part 5, Trigonometry
Lecture 5.1b, Applications of the Sum and Difference Formulas
Dr. Ken W. Smith
Sam Houston State University
2013
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Elementary Functions
2013
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Elementary Functions
A reduction formula
Sum and difference formulae for sine and cosine
By our sum-of-angles formulas,
sin(x + θ) = sin x cos θ + cos x sin θ.
If we replace cos θ and sin θ by
√ a
a2 +b2
a sin x + b cos x =
√ b
, we get
a2 +b2
√x+b cos x .
= a sin
a2 +b2
√
a2 + b2 sin(x + θ).
and sin θ =
sin(x + θ) = sin x √a2a+b2 + cos x √a2b+b2
√
We clear the denominators by multiplying all sides by a2 + b2 .
√
a sin x + b cos x = a2 + b2 sin(x + θ).
(3)
This formula is useful for changing a linear combination of sine and cosine
functions into just a sine function.
√
For example, the function f (x) = sin x + 3 cos x can be rewritten as
q
√
√ 2
sin x + 3 cos x = 12 + 3 sin(x + θ)
where θ is√the angle between the x-axis and the line from the origin to the
point
q (1, 3). Since θ = π/3 in this problem, and since
√ 2 √
12 + 3 = 1 + 3 = 2, we have
sin x +
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Sum and difference formulas for tangent
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sin(α+β)
cos(α+β)
=
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sin α cos β+cos α sin β
cos α cos β−sin α sin β
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tan α + tan β
.
1 − tan α tan β
What if we wanted an equation for tan(α − β)?
tan α + tan β
.
1 − tan α tan β
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tan(α + β) =
But we would really like a sum-of-angles formula for tangent that is in
terms of tan α and tan β.
So let’s divide both numerator and denominator by cos α cos β.
tan(α + β) =
3 cos x = 2 sin(x + π/3).
Sum and difference formulae for tangent
sin(α+β)
We know that tan(α + β) = cos(α+β)
so we may use our sum-of-angle
formulas to create a formula for tan(α + β).
A first pass gives
tan(α + β) =
√
Replace β by −β and use the fact that tangent is an odd function to
obtain
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tan(α − β) =
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tan α − tan β
.
1 + tan α tan β
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Some worked problems
Sum and difference formulae for sine and cosine
Compute the exact value of cos 75◦ .
Compute the exact value of sin 15◦ .
Solution.
Using the sum of angles formula for cosine and the fact that
75◦ = 45◦ + 30◦ , we have
Solution.
sin 15◦ = sin(45◦ − 30◦ ) = sin 45◦ cos 30◦ − cos 45◦ sin 30◦
√ √
√
√
√
2 3
21
6− 2
=
−
=
2 2
2 2
4
cos 75◦ = cos(45◦ + 30◦ ) = cos 45◦ cos 30◦ − sin 45◦ sin 30◦
√
√ √
√
√
2 3
21
6− 2
=
−
=
2 2
2 2
4
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This answer looks familiar! (Notice that since 75◦ and 15◦ are
complementary angles so the cosine of one angle is the sine of the other!)
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Sum and difference formulae for sine and cosine
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Sum and Difference Formulas
Find the tangent of 75◦ .
Solution. We could separately compute the sine and cosine of 75◦ using
our sum-of-angle formulas (as we did on the last slides) or we could use
the sum-of-angles formula for tangent and write
√
3
1+
◦ + tan 30◦
tan
45
√3
tan 75◦ = tan(45◦ + 30◦ ) =
=
.
1 − tan 30◦ tan 45◦
3
1−(
)(1)
3
Multiplying numerator and denominator by 3 gives
√
3+ 3
◦
√ .
tan 75 =
3− 3
If we don’t like the square roots in the
√ denominator, we can multiply
numerator and denominator by 3 + 3 (the conjugate of the denominator)
and find that
√
√
√
√
√
3+ 3 3+ 3
9+6 3+3
12 + 6 3
◦
√ )=
tan 75Smith
= ((SHSU) √ )(
(
)=(
) =20132 + 27 /328.
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Functions
9−3
6
In the next presentation, we will look at double angle and power reduction
formulas.
(End)
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