Extensions
Special angles
Symmetry
Identities
Elementary graphs
Trigonometry, Part II
Tom Lewis
Spring Semester
2014
Extensions
Special angles
Symmetry
Outline
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Identities
Elementary graphs
Extensions
Special angles
Symmetry
Identities
Elementary graphs
An observation
Let α be a positive angle less than π/2 radians and in standard
position. Let a circle of radius r be centered at the origin. Let
P (x , y) be the point at which the terminal ray of the angle
intersects the circle and notice that
cos(α) =
Extensions
x
r
Special angles
and
sin(α) =
Symmetry
y
.
r
Identities
Elementary graphs
Definition (The trigonometric functions, extended)
Let α be a given directed angle in standard position and let a circle
of radius r be centered at the origin. Let P (x , y) be the point of
intersection of the terminal ray of angle α and the circle. Then
cos(α) =
x
r
and
sin(α) =
y
.
r
In particular, this means that the point P has coordinates
(r cos(α), r sin(α)).
The remaining four trigonometric functions are defined by
tan(α) =
sin(α)
cos(α)
sec(α) =
1
cos(α)
cot(α) =
cos(α)
sin(α)
csc(α) =
1
sin(α)
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Problem
For which angles α ∈ [0, 2π] is sin(α) = 0 or sin(α) = 1? Repeat
this problem for the cosine function.
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Problem
The terminal side of a directed angle α passes through the point
P (5, −12). What are the values of the 6 trig. functions at this
angle?
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Theorem (Periodicity of sine and cosine)
For any angle α (in radian measure),
sin(α + 2π) = sin(α) and
cos(α + 2π) = cos(α).
Corollary
The remaining 4 trigonometric functions are periodic with period
2π as well.
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Elementary angles
Using the 30-60-right (π/6 − π/3 − π/2) and the 45-45-right
(π/4 − π/4 − π/2) elementary triangles and symmetry, it is very
easy to obtain the values of the sine and cosine functions at all of
the elementary angles between 0 and 2π; see the table on page A4.
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Symmetry
Identities
Elementary graphs
Problem
Evaluate the following:
1. cos(7π/6)
2. sin(−2π/3)
3. tan(11π/4)
Extensions
Special angles
Problem
1. Solve the equation sec(x ) = 2 for x ∈ [0, 2π].
2. Solve the equation 2 sin2 (x ) + sin(x ) = 1 for x ∈ [0, 2π].
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Theorem
The cosine function is even and the sine function is odd, that is, for
any angle α,
cos(−α) = cos(α) and
sin(−α) = − sin(α).
Problem
Do the remaining 4 trigonometric functions have an even or odd
symmetry?
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Theorem (The Pythagorean Identity)
For any angle α,
sin2 (α) + cos2 (α) = 1.
Other identities
From this identity we can discover two additional important
identities:
tan2 (α) + 1 = sec2 (α)
and
1 + cot2 (α) = csc2 (α).
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Problem
Suppose that sin(α) = −.6 and π < α < 3π/2. Find the values of
the other trig functions at α.
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Problem
Suppose that tan(β) = .75. What are the other possible values of
cos(β)?
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Theorem (Expansion Formulas)
For any angles α and β,
sin(α + β) = sin(α) cos(β) + sin(β) cos(α)
cos(α + β) = cos(α) cos(β) − sin(α) sin(β)
Likewise
sin(α − β) = sin(α) cos(β) − sin(β) cos(α)
cos(α − β) = cos(α) cos(β) + sin(α) sin(β)
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Problem
1. Find the exact value of sin(π/12).
2. Find the exact value of cos(7π/12).
3. Show that sin(α + π/2) = cos(α).
4. Show that tan(α + π) = tan(α). This shows that tan has
period π.
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Theorem (Double-angle formulas)
For any angle x ,
sin(2x ) = 2 sin(x ) cos(x )
cos(2x ) = cos2 (x ) − sin2 (x )
cos(2x ) = 2 cos2 (x ) − 1
cos(2x ) = 1 − 2 sin2 (x )
Extensions
Special angles
Symmetry
Identities
Theorem (Half-angle formulas)
For an angle x ,
1 + cos(2x )
2
1 − cos(2x )
sin2 (x ) =
2
cos2 (x ) =
Elementary graphs
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Theorem (Product formulas)
For any angles x and y,
1
sin(x + y) + sin(x − y)
2
1
cos(x + y) + cos(x − y)
cos(x ) cos(y) =
2
1
sin(x ) sin(y) =
cos(x − y) − cos(x + y)
2
sin(x ) cos(y) =
Extensions
Special angles
Symmetry
Identities
Problem
1. Show that tan2 (α) − sin2 (α) = tan2 (α) sin2 (α).
2. Show that sin2 (x ) − sin2 (y) = sin(x + y) sin(x − y).
Elementary graphs
Extensions
Special angles
Symmetry
Identities
Elementary graphs
Problem
Develop the graphs of y = sin(x ) and y = cos(x ) on the interval
[0, 2π]. From these, develop the graphs of the remaining
trigonometric functions.
Extensions
Special angles
Symmetry
Identities
Elementary graphs
3
2
1
Π
2
Π
3Π
2
-1
-2
-3
Figure : Sine (blue) and cosecant (light blue)
2Π
Extensions
Special angles
Symmetry
Identities
Elementary graphs
3
2
1
Π
Π
2
3Π
2Π
2
-1
-2
-3
Figure : Cosine (red) and cosecant (light red)
Extensions
Special angles
Symmetry
Identities
Elementary graphs
3
2
1
Π
2
Π
3Π
2
-1
-2
-3
Figure : Sine (blue), cosine (red), and tangent (green)
2Π
Extensions
Special angles
Symmetry
Identities
Elementary graphs
3
2
1
Π
Π
2
3Π
2Π
2
-1
-2
-3
Figure : Sine (blue), cosine (red), and cotangent (light green)
Extensions
Special angles
Symmetry
Problem
1. Graph y = 2 − cos(x ).
2. Graph y = 3 cos(x − π/3).
Identities
Elementary graphs