sum and di erence formulae for sine and cosinekws006/precalculus/5.1_sum_and...part 5, trigonometry...
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Elementary FunctionsPart 5, Trigonometry
Lecture 5.1a, Sum and Difference Formulas
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU) Elementary Functions 2013 1 / 28
Sum and difference formulae for sine and cosine
Consider angles α and β with α > β. These angles identify points on theunit circle, P (cosα, sinα) and Q(cosβ, sinβ).
Smith (SHSU) Elementary Functions 2013 2 / 28
Sum and difference formulae for sine and cosine
The distance between P and Q is, by the Pythagorean theorem, thesquare root of
d(PQ)2 = (cosα− cosβ)2 + (sinα− sinβ)2.
Smith (SHSU) Elementary Functions 2013 3 / 28
Sum and difference formulae for sine and cosine
We can expand this out, algebraically, and get
d(PQ)2 = (cos2 α−2 cosα cosβ+cos2 β)+(sin2 α−2 sinα sinβ+sin2 β).
which we can rewrite (using the Pythagorean identity) as
d(PQ)2 = 2− 2 cosα cosβ − 2 sinα sinβ
Smith (SHSU) Elementary Functions 2013 4 / 28
Sum and difference formulae for sine and cosine
Let’s rotate this picture clockwise through the angle −β so that the pointQ becomes Q′(1, 0) lying on the x-axis. The point P rotates into thepoint P ′(cos(α− β), sin(α− β))The distance from P ′ to Q′ is the same as the distance from P to Q.
Smith (SHSU) Elementary Functions 2013 5 / 28
Sum and difference formulae for sine and cosine
Smith (SHSU) Elementary Functions 2013 6 / 28
Sum and difference formulae for sine and cosine
The distance from P ′ to Q′ is the square root of
d(P ′Q′)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).
Smith (SHSU) Elementary Functions 2013 7 / 28
Sum and difference formulae for sine and cosine
Since the line segments PQ and P ′Q′ are congruent, then we know thatd(PQ)2 = d(P ′Q′)2.
Smith (SHSU) Elementary Functions 2013 8 / 28
Sum and difference formulae for sine and cosine
In mathematics, if we arrive at the same value through two differentcomputations, we always have valuable information. Here we can equated(PQ)2 and d(P ′Q′)2 and simplify.
Smith (SHSU) Elementary Functions 2013 9 / 28
Sum and difference formulae for sine and cosine
D2 = 2− 2 cosα cosβ − 2 sinα sinβ
D2 = 2− 2 cos(α− β)2Smith (SHSU) Elementary Functions 2013 10 / 28
Sum and difference formulae for sine and cosine
So2− 2 cosα cosβ − 2 sinα sinβ = 2− 2 cos(α− β)2.
We may divide both sides by 2 and solve for cos(α− β).
cos(α− β) = cosα cosβ + sinα sinβ
This is an important result.
Smith (SHSU) Elementary Functions 2013 11 / 28
Sum and difference formulae for sine and cosine
We create a more memorable form of this equation if we replace β by −βand use the fact that sine is an odd function while cosine is an evenfunction.
Write cos(α+ β) = cos(α− (−β)) and replace β by −β in our equationfor cos(α− β) to see that
cos(α+ β) = cosα cos(−β) + sinα sin(−β) = cosα cos(β)− sinα sin(β)
This is an equation we want to record and use on a regular basis.
cos(α+ β) = cosα cosβ − sinα sinβ (1)
Smith (SHSU) Elementary Functions 2013 12 / 28
Sum and difference formulae for sine and cosine
If we note that sin(θ) = cos(π
2− θ) (since sine and cosine are
complementary functions!) then we can write
sin(α+ β) = cos(π
2− (α+ β)) = cos((
π
2− α)− β).
Using the sum-of-angles formula above, with regards to the anglesπ
2− α
and β, we have
sin(α+ β) = cos((π
2− α)− β)
= cos(π
2− α) cosβ + sin(
π
2− α) sinβ = sinα cosβ + cosα sinβ.
Smith (SHSU) Elementary Functions 2013 13 / 28
Sum and difference formulae for sine and cosine
Therefore we have a sum-of-angles equation for the sine function:
sin(α+ β) = sinα cosβ + cosα sinβ (2)
If we need, we may replace β by −β to create a difference equation:
sin(α− β) = sinα cosβ − cosα sinβ
Smith (SHSU) Elementary Functions 2013 14 / 28
Sum and difference formulae for sine and cosine
I don’t memorize these identities.
(That’s one reason I’ve been attracted to mathematics – if oneunderstands the math, one doesn’t need to memorize!)
In the undergraduate classes that I teach at Sam Houston State University,I will provide students with the sum of angle formulas when they areneeded.
Smith (SHSU) Elementary Functions 2013 15 / 28
Sum and difference formulae for sine and cosine
Here they are again:
cos(α+ β) = cosα cosβ − sinα sinβ
sin(α+ β) = sinα cosβ + cosα sinβ
Smith (SHSU) Elementary Functions 2013 16 / 28
Sum and difference formulae for sine and cosine
Here they are as difference of angles
cos(α−β) = cosα cosβ+sinα sinβ
sin(α−β) = sinα cosβ− cosα sinβ
Smith (SHSU) Elementary Functions 2013 17 / 28
Sum and Difference Formulas
In the next presentation, we will look at some applications of these sumand difference formulas.
(End)
Smith (SHSU) Elementary Functions 2013 18 / 28
Elementary FunctionsPart 5, Trigonometry
Lecture 5.1b, Applications of the Sum and Difference Formulas
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU) Elementary Functions 2013 19 / 28
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 = a√a2+b2
, sin θ = b√a2+b2
.
Smith (SHSU) Elementary Functions 2013 20 / 28
A reduction formula
By our sum-of-angles formulas,
sin(x+ θ) = sinx cos θ + cosx sin θ.
If we replace cos θ and sin θ by a√a2+b2
and sin θ = b√a2+b2
, we get
sin(x+ θ) = sinx a√a2+b2
+ cosx b√a2+b2
= a sinx+b cosx√a2+b2
.
We clear the denominators by multiplying all sides by√a2 + b2.
a sinx+ b cosx =√a2 + b2 sin(x+ θ). (3)
Smith (SHSU) Elementary Functions 2013 21 / 28
Sum and difference formulae for sine and cosine
a sinx+ b cosx =√a2 + b2 sin(x+ θ).
This formula is useful for changing a linear combination of sine and cosinefunctions into just a sine function.For example, the function f(x) = sinx+
√3 cosx can be rewritten as
sinx+√3 cosx =
√12 +
√32sin(x+ θ)
where θ is the angle between the x-axis and the line from the origin to thepoint (1,
√3). Since θ = π/3 in this problem, and since√
12 +√32=√1 + 3 = 2, we have
sinx+√3 cosx = 2 sin(x+ π/3).
Smith (SHSU) Elementary Functions 2013 22 / 28
Sum and difference formulas for tangent
We know that tan(α+ β) = sin(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) = sin(α+β)cos(α+β) =
sinα cosβ+cosα sinβcosα cosβ−sinα sinβ
But we would really like a sum-of-angles formula for tangent that is interms of tanα and tanβ.So let’s divide both numerator and denominator by cosα cosβ.
tan(α+ β) =tanα+ tanβ
1− tanα tanβ. (4)
Smith (SHSU) Elementary Functions 2013 23 / 28
Sum and difference formulae for tangent
tan(α+ β) =tanα+ tanβ
1− tanα tanβ.
What if we wanted an equation for tan(α− β)?
Replace β by −β and use the fact that tangent is an odd function toobtain
tan(α− β) = tanα− tanβ
1 + tanα tanβ.
Smith (SHSU) Elementary Functions 2013 24 / 28
Some worked problems
Compute the exact value of cos 75◦.
Solution.Using the sum of angles formula for cosine and the fact that75◦ = 45◦ + 30◦, we have
cos 75◦ = cos(45◦ + 30◦) = cos 45◦ cos 30◦ − sin 45◦ sin 30◦
=
√2
2
√3
2−√2
2
1
2=
√6−√2
4
Smith (SHSU) Elementary Functions 2013 25 / 28
Sum and difference formulae for sine and cosine
Compute the exact value of sin 15◦.
Solution.
sin 15◦ = sin(45◦ − 30◦) = sin 45◦ cos 30◦ − cos 45◦ sin 30◦
=
√2
2
√3
2−√2
2
1
2=
√6−√2
4
This answer looks familiar! (Notice that since 75◦ and 15◦ arecomplementary angles so the cosine of one angle is the sine of the other!)
Smith (SHSU) Elementary Functions 2013 26 / 28
Sum and difference formulae for sine and cosine
Find the tangent of 75◦.
Solution. We could separately compute the sine and cosine of 75◦ usingour sum-of-angle formulas (as we did on the last slides) or we could usethe sum-of-angles formula for tangent and write
tan 75◦ = tan(45◦ + 30◦) =tan 45◦ + tan 30◦
1− tan 30◦ tan 45◦=
1 +
√3
3
1− (
√3
3)(1)
.
Multiplying numerator and denominator by 3 gives
tan 75◦ =3 +√3
3−√3.
If we don’t like the square roots in the denominator, we can multiplynumerator and denominator by 3 +
√3 (the conjugate of the denominator)
and find that
tan 75◦ = (3 +√3
3−√3)(3 +√3
3 +√3) = (
9 + 6√3 + 3
9− 3) = (
12 + 6√3
6) = 2 +
√3 .Smith (SHSU) Elementary Functions 2013 27 / 28
Sum and Difference Formulas
In the next presentation, we will look at double angle and power reductionformulas.
(End)
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