Download Lecture 1: Trigonometric Functions: Definitions

Lecture 1: Trigonometric Functions: Definitions
1.1 The sine, cosine, and tangent functions
We say triangles ∆ABC and ∆DEF are similar if 6 A = 6 D, 6 B = 6 E, and 6 C = 6 F .
An important fact about similar triangles is that the ratios of corresponding sides are
equal. That is,
AB
DE
=
,
AC
DF
AB
DE
=
,
BC
EF
and
AC
DF
=
.
BC
EF
Note that two right triangles are similar if they have a pair of equal acute angles (the
remaining two angles are then equal because one is a right angle and the sum of the angles
of triangle must equal 180◦ ). It follows that the ratios of the sides of a right triangle are
determined completely by either one of its acute angles. This is the basis for the following
definition.
Definition Suppose ∆ABC is a right triangle with right angle at A. Let θ be the angle
at C. Then the sine of θ is
AB
sin(θ) =
,
BC
the cosine of θ is
AC
cos(θ) =
,
BC
and tangent of θ is
AB
.
tan(θ) =
AC
Note that
tan(θ) =
sin(θ)
.
cos(θ)
Example Suppose 4ABC is a right triangle with right angle at A and acute angles
◦
which are both
√ 45 . If the legs AB and AC are each of length 1, then the hypotenuse BC
is of length 2. Hence
1
sin(45◦ ) = √ ,
2
1
cos(45◦ ) = √ ,
2
and
tan(45◦ ) = 1.
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Lecture 1: Trigonometric Functions: Definitions
1-2
Example Suppose ∆ABC is an equilateral triangle with each side of length 1. Let D
be the base of the perpendicular from B to the side AC. Then ∆DBC is a right triangle
with acute angle 6 B = 30◦ and acute angle 6 C = 60◦ . Now BC has length 1, DC has
length 12 , and BD has length
r
1
1− =
4
r
√
3
3
=
.
4
2
Hence
√
3
,
2
1
cos(60◦ ) = ,
2
sin(60◦ ) =
and
tan(60◦ ) =
We also see that
√
3.
1
,
2
√
3
◦
cos(30 ) =
,
2
sin(30◦ ) =
and
1
tan(30◦ ) = √ .
3
We complete the definition of sine, cosine, and tangent for all angles between 0◦ and 90◦
with the following:
sin(0◦ ) = 0,
cos(0◦ ) = 1,
tan(0◦ ) = 0,
sin(90◦ ) = 1,
and
cos(90◦ ) = 0.
Note that tangent is not defined for 90◦ .
1.2 Extending the definitions
We now extend the definitions of sine, cosine, and tangent to functions on all of (−∞, ∞).
To do so, we first note that we may restrict our discussion to right triangles with hypotenuses of length 1. In that case, if we place one acute angle at the origin and the
right angle on the x-axis, the other vertex of the triangle will lie on the unit circle C with
Lecture 1: Trigonometric Functions: Definitions
1-3
equation x2 + y 2 = 1. In fact, choosing any point on C determines such a triangle. If (a, b)
is a point on C, the acute angle θ determined by (a, b) is the angle between the line from
(a, b) to the origin and the x-axis. Instead of measuring this with degrees, we will now
measure the angle by the distance from (1, 0) to (a, b) measured along the circle in the
counterclockwise direction. We can then allow for θ < 0, meaning that (a, b) is a distance
|θ| from (1, 0) in the clockwise direction, and we may allow for |θ| ≥ 2π, meaning that we
move around the circle one or more times before reaching (a, b). With these conventions,
if we move a distance θ from (1, 0) to a point (a, b) on the unit circle C, we define
sin(θ) = b,
cos(θ) = a,
and
tan(θ) =
b
.
a
When measuring angles in this way, the units of measurement are called radians. Note
that, since the circle has a circumference of 2π, we have
90◦ =
2π
π
radians = radians.
4
2
Hence
1◦ =
and
Example
π
radians
180
180 ◦
1 radian =
.
π
For our standard angles we have
π
radians,
6
π
45◦ = radians,
4
30◦ =
and
π
radians.
3
Hence we may now restate our previous values of sine, cosine, and tangent:
π √3
π
π 1
π
1
sin
=
sin
= 1.
sin(0) = 0
sin
=
sin
=√
3
2
2
6
2
4
2
π √3
π
π 1
π
1
cos(0) = 1
cos
=
cos
=√
cos
=
cos
= 0.
6
2
4
3
2
2
2
π
π
π √
1
tan(0) = 0
tan
=√
tan
=1
tan
= 3.
6
4
3
3
60◦ =
Lecture 1: Trigonometric Functions: Definitions
1-4
Example Using the basic values from the previous example and the geometry of the
circle, we may now find values for sine, cosine, and tangent for many different angles. For
example, we have
sin(π) = 0,
cos(π) = −1,
and
tan(π) = 0.
Example
We have
sin
and
cos
Note that tangent is not defined at
Example
3π
2
3π
2
= −1,
= 0.
3π
2 .
We have
sin(2π) = 0,
cos(2π) = 1,
and
tan(2π) = 0.
Example
We have
sin(3π) = 0,
cos(3π) = −1,
and
tan(3π) = 0.
Example
We have
5π
1
sin
= ,
6
2
√
5π
3
cos
=−
,
6
2
5π
1
tan
= −√ .
6
3
Lecture 1: Trigonometric Functions: Definitions
Example
We have
5π
1
sin −
=− ,
6
2
√
5π
3
cos −
=−
,
6
2
5π
1
tan −
=√ .
6
3
1.3 The other trigonometric functions
We may now define the remaining three trigonometric functions:
sec(θ) =
1
,
cos(θ)
csc(θ) =
1
,
sin(θ)
and
cot(θ) =
Example
and
1
cos(θ)
=
.
tan(θ)
sin(θ)
We have
sec
2π
3
=
csc
2π
3
=
1
1
=
= −2,
1
2π
−
cos
2
3
1
1
2
= √ =√ ,
2π
3
3
sin
3
2
2π
1
cos
−
1
2π
3
= √2 = − √ .
cot
=
2π
3
3
3
sin
3
2
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