CRASH COURSE IN PRECALCULUS
Shiah-Sen Wang
The graphs are prepared by Chien-Lun Lai
Based on : Precalculus: Mathematics for Calculus
by J. Stuwart, L. Redin & S. Watson,
6th edition, 2012, Brooks/Cole
Chapter 5-7.
LECTURE 7. TRIGONOMETRY: PART II
This lecture is the second part of reviewing high school
trigonometry: addition and subtraction , double and half angle,
product-to-sum formulas and sum-to-product formulas for
trigonometric functions and some of their applications; and
area formulas and the laws of sines and cosines for general
triangles.
Addition and Subtraction Formulas for Trig. Function
⎧
⎪
⎪ sin(α + β) = sin α cos β + cos α sin β
Formulas for Sine: ⎨
⎪
⎪
⎩ sin(α − β) = sin α cos β − cos α sin β
⎧
⎪
⎪ cos(α + β) = cos α cos β − sin α sin β
Formulas for Cosine: ⎨
⎪
⎪
⎩ cos(α − β) = cos α cos β + sin α sin β
⎧
tan α + tan β
⎪
⎪
tan(α + β) =
⎪
⎪
⎪
1 − tan α tan β
Formulas for Tangent: ⎨
tan α − tan β
⎪
⎪
tan(α − β) =
⎪
⎪
⎪
1 + tan α tan β
⎩
Some Proofs of the Formulas
Proof of Subtraction Formula for Sine:
By the addition formula for sine and even-odd identities,
sin(α − β) = sin [α + (−β)] = sin α cos(−β) + cos α sin(−β) =
sin α cos β − cos α sin β.
Proof of Subtraction Formula for Cosine:
By the addition formula for cosine and even-odd identities,
cos(α − β) = cos [α + (−β)] = cos α cos(−β) − sin α sin(−β) =
cos α cos β + sin α sin β.
Proof Addition Formula for Tangent:
By Addition Formula for Sine and Cosine and Reciprocal
Identities,
sin(α + β) sin α cos β + cos α sin β
=
=
tan(α + β) =
cos(α + β) cos α cos β − sin α sin β
sin α cos β+cos α sin β
cos α cos β
cos α cos β−sin α sin β
cos α cos β
=
tan α + tan β
1 − tan α tan β
Examples
7π
=?
12
7π
3π + 4π
π π
sin
= sin (
) = sin ( + ) =
12
12
√ 4 3√ √
√ √
π
π
π
21
2 3 (1 + 3) 2
π
+
=
.
sin cos + cos sin =
4
3
4
3
2 2
2 2
4
π
2π
π
2π
2. cos cos
− sin sin
=?
9
9
9
9
π
2π
π
2π
π 2π
π 1
cos cos
− sin sin
= cos ( +
) = cos = .
9
9
9
9
9
9
3 2
1 + tan θ
π
3. Prove the identity
= tan ( + θ).
1 − tan θ
4
π
tan 4 + tan θ
1 + tan θ
π
=
= tan ( + θ).
1 − tan θ 1 − tan π4 tan θ
4
1. sin
Examples
4. Express sin(cos−1 x + tan−1 y) as an algebraic expression
in x and y, where x ∈ [−1, 1] and y ∈ R.
Let α = cos−1 x and β = tan−1 y. Then cos α = x, tan β = y
and
sin(cos−1 x + tan−1 y)
√ = sin(α + β) = sin α cos β + cos α sin β.
cos α = x ⇒ sin α = 1 − x 2 .
tan β = y ⇒ sin β = √ y 2 and cos β = √ 1 2 . Hence
1+y
1+y
√
1
y
sin(cos−1 x + tan−1 y) = 1 − x 2 √
+x√
=
1 + y2
1 + y2
√
√
√
2
2
1 − x 2 + xy ( 1 − x + xy) 1 + y
√
=
.
1 + y2
1 + y2
Examples
5. Use the addition
and subtraction for sine to simplify
√
1
3
sin θ +
cos θ in terms of a single trigonometric
2
2
function. √
3
1
π
π
sin θ +
cos θ = cos sin θ + sin cos θ = sin (θ +
2
2
3
3
On the other
√ hand,
1
3
π
π
sin θ +
cos θ = sin sin θ + cos cos θ = sin (θ −
2
2
6
6
π
).
3
π
).
6
More generally, we can write, for any A, B ∈ R with A2 + B 2 ≠ 0,
√
B
cos θ + √
sin θ)
2
A + B2
√
√
= A2 + B 2 (cos ϕ1 cos θ + sin ϕ1 sin ϕ1 ) = A2 + B 2 cos(θ − ϕ1 )
√
√
= A2 + B 2 (sin ϕ2 cos θ + cos ϕ2 sin ϕ2 ) = A2 + B 2 sin(θ + ϕ2 )
A cos θ + B sin θ =
A2 + B 2 ⋅ ( √
A
A2
+ B2
Sum of Sines and Cosines
For any A, B ∈ R with A2 + B 2 ≠ 0
√
√
A cos θ + B sin θ = A2 + B 2 cos(θ − ϕ1 ) = A2 + B 2 sin(θ + ϕ2 )
⎧
A
⎪
⎪cos ϕ1 = √A2 +B 2
where ⎨
⎪
√ B
⎪
⎩sin ϕ1 = A2 +B 2
and
⎧
A
⎪
⎪sin ϕ2 = √A2 +B 2
⎨
⎪
√ B
⎪
⎩cos ϕ2 = A2 +B 2
We note that by knowing the values any two distinct
trigonometric functions of the six trigonometric functions, as
long as they are not from the three reciprocal identities, the
values of remaining four trigonometric functions are also
determined.
An Example
Prove the identity
π
cos θ − sin θ
= tan ( − θ).
cos θ + sin θ
4
√ √2
2 ( 2 cos θ −
cos θ − sin θ
=√ √
cos θ + sin θ
2 ( 22 cos θ +
=
sin π4 cos θ − cos
cos π4 cos θ + sin
π
4
π
4
sin θ
=
sin θ
√
√
√
2
2
2
cos
θ
−
2 sin θ)
2
√
√
= √2
2
2
2
2 sin θ)
2 cos θ + 2
sin ( π4 − θ)
π
= tan ( − θ)
π
4
cos ( 4 − θ)
sin θ
sin θ
Double Angle Formulas for Trigonometric Functions
With α = β = θ in the addition formulas for sine, cosine and
tangent functions, we have
Formula for Sine Function:
sin 2θ = 2 sin θ cos θ
Formula for Cosine Function:
⎧
cos2 θ − sin2 θ
⎪
⎪
⎪
⎪
cos 2θ = ⎨1 − 2 sin2 θ
⎪
⎪
⎪
2
⎪
⎩2 cos θ − 1
Formula for Tangent Function:
tan 2θ =
2 tan θ
.
1 − tan2 θ
Half Angle Formula for Sine and Cosine Functions
⎧
2
⎪
θ
⎪1 − 2 sin θ
Using cos 2θ = ⎨
, replacing 2θ, θ by θ,
2
⎪
2
⎪
⎩2 cos θ − 1
correspondingly and rearrange terms, we obtain
sin2
θ 1 − cos θ
=
,
2
2
cos2
θ 1 + cos θ
=
2
2
and so
√
θ
1 − cos θ
sin = ±
,
2
2
√
θ
1 + cos θ
cos = ±
2
2
The choice of + or − sign depends on the quadrant in which
lies.
θ
2
Half Angle Formula for Tangent Function
θ
2 tan 2
2 tan θ
=
Using tan 2θ =
⇒
tan
θ
=
1 − tan2 θ
1 − tan2 2θ
2t
θ
, where t = tan , and
2
1−t
2
√
2
we
choose
b
=
2t
and
a
=
1
−
t
,
so
c
=
(2t)2 + (1 − t 2 )2 =
√
√
√
2
2
4
2
4
4t + 1 − 2t + t = 1 + 2t + t = (1 + t 2 )2 = 1 + t 2 .
Therefore,
Half Angle Formula for Tangent Function
sin θ =
cos θ =
tan θ =
2 tan 2θ
1 + tan2
θ
2
1 − tan2
θ
2
1 + tan2 2θ
2 tan 2θ
1 − tan2
θ
2
csc θ =
sec θ =
cot θ =
1 + tan2
θ
2
2 tan 2θ
1 + tan2
θ
2
1 − tan2 2θ
1 − tan2
2 tan 2θ
θ
2
Examples
1. sin
π
=?
12
π
Note that sin
we see that
π
π
= sin 6 and
lies in the first quadrant,
12
2
12
√
sin
π
=
12
1 − cos
π
6
¿
√
√
Á1 − 3 √
Á
À
2− 3
2
=
=
2
2
2
7π
2. tan
=?
8
7π
π 3π
π
3π
3π
tan
= tan ( +
) = tan [ − (− )] = cot (− ) =
8
2
8
2
8
8
3π
3π
cos
3π
3π
8
− cot
=−
and
= 4 is in the the first quadrant.
3π
8
8
2
sin 8
√
¿
¿
√
1+cos 3π
√
4
Á1 − 2
Á2 − 2
7π
Á
2
Á
À
2
À
√ =−
√ =
Hence tan
= −√
= −Á
8
2+ 2
1−cos 3π
1 + 22
4
2
√
2− 2
Examples
3. Write sin(2 cos−1 x) as an algebraic expression in x only,
where x ∈ [−1, 1].
Let θ = cos−1 x. Then cos θ = x and
sin(2 cos−1 x) = sin 2θ = 2 sin θ cos θ. Using
√
we have sin θ = 1 − x 2 . √
Hence sin(2 cos−1 x) = 2x 1 − x 2 .
Product-to-Sum Formulas
⎧
⎪
⎪ sin(α + β) = sin α cos β + cos α sin β
Idea: Recalling ⎨
and
⎪
sin(α
−
β)
=
sin
α
cos
β
−
cos
α
sin
β
⎪
⎩
adding the left- and right-sides of these formulas, gives
sin(α + β) + sin(α − β) = 2 sin α cos β
So
sin(α + β) + sin(α − β)
2
Similarly, by subtracting them on both sides, gives
sin α cos β =
cos α sin β =
sin(α + β) − sin(α − β)
2
Apply similar techniques to the addition and subtraction
formulas of cosine function to see
Product-to-Sum Formulas
sin(α + β) + sin(α − β)
2
sin(α + β) − sin(α − β)
cos α sin β =
2
cos(α + β) + cos(α − β)
cos α cos β =
2
cos(α − β) − sin(α + β)
sin α sin β =
2
sin α cos β =
Examples
1. Express sin 3x cos 5x as a sum of trigonometric functions.
sin(3x + 5x) + sin(3x − 5x)
2
sin 8x + sin(−2x) sin 8x − sin 2x
=
or
=
2
2
sin 3x cos 5x = cos 5x sin 3x
sin(5x + 3x) − sin(5x − 3x) sin 8x − sin 2x
=
=
2
2
2. Express sin 3x sin 5x as a sum of trigonometric functions.
sin 3x cos 5x =
cos(3x − 5x) − cos(3x − 5x)
2
cos(−2x) − cos 8x cos 2x − cos 8x
=
=
2
2
sin 3x sin 5x =
Sum-to-Product Formulas
By applying the Product-to-Sum formulas, with
⎧
⎧
x+y
⎪
⎪
⎪α = 2
⎪x = α + β
, and rearranging terms, we obtain
⎨
⇒⎨
x−y
⎪
⎪
⎪
⎪
⎩β = 2
⎩y = α − β
sin x + sin y = 2 sin
x +y
x −y
cos
2
2
x +y
x −y
sin
2
2
x +y
x −y
cos x + cos y = 2 cos
cos
2
2
x −y
x +y
sin
cos x − cos y = −2 sin
2
2
sin x − sin y = 2 cos
Examples
1. Write sin 7x + sin 3x as a multiple of trigonometric
functions.
7x + 3x
7x − 3x
sin 7x + sin 3x = 2 sin
cos
= 2 sin 5x cos 2x.
2
2
sin 3x − sin x
2. Simply the fractional expression
.
cos 3x + cos x
3x+x
3x−x
2 cos 2 sin 2
sin 3x − sin x
2 cos 2x sin x
=
=
=
3x+x
3x−x
cos 3x + cos x 2 cos 2 cos 2
2 cos 2x cos x
sin x
= tan x.
cos x
Area Formulas for General Triangles
Given a triangle with side lengths a and b, and included angle
θ, then the area
1
A = ab sin θ
2
Area Formulas for General Triangles
Heron’s Formula:
√
A=
s(s − a)(s − b)(s − c)
a+b+c
is the semiperimeter of the triangle; that is
2
half of the perimeter.
where s =
Laws of Sines and Cosines for General Triangles
Law of Sines: In ∆ABC we have
sin α sin β sin γ
=
=
a
b
c
Law of Cosines: In ∆ABC
a2 = b2 + c 2 − 2bc cos α
b2 = a2 + c 2 − 2ac cos β
c 2 = a2 + b2 − 2ab cos γ