Transcript Circular functions

Unit Circle Definition of Trig Functions
The Unit Circle
A unit circle is the circle with center at the
origin and radius equal to 1 (one unit).
 Its equation is (x – 0)2 + (y – 0)2 = 1 or
x2 + y2 = 1.

Coordinate Function
The coordinate function P is the function from
the set of P(θ) = {(x, y) | x2 + y2 = 1} to the set
of real numbers R.
 Examples:

1.
P(
3π
2
) is the terminal point of the arc length
so that P(
3π
2
) = (0, -1).
If θ = -π, then P(-π) = (-1, 0).
3. If θ = 0, then P(0) = (1, 0).
2.
4.
If θ =
5.
If θ =
5π
5π
, then P( ) = (0, 1).
2
2
−3π
−3π
, then P(
) = (0, 1).
2
2
Note: The given θ are quadrantal angles.
3π
2
0, π/2, π, 3π/2, and 2π (and their multiples)
are called quadrantal angles.
 Note that P(θ) = (x, y).

Circular Functions

Trigonometric functions are defined so
that their domains are sets of
angles and their ranges are sets of real
numbers. Circular functions are
defined such that their domains are sets
of numbers that correspond to the
measures (in radian units) of the angles
of analogous trigonometric functions.
Circular Functions
The ranges of these circular functions,
like their analogous trigonometric
functions, are sets of real numbers.
 These functions are called circular
functions because radian measures of
angles are determined by the lengths of
arcs
of
circles.
In
particular,
trigonometric functions defined using the
unit circle lead directly to these circular
functions.

Circular Functions

The circle below is drawn in a coordinate system where the
circle's center is at the origin and has a radius of 1. This circle
is known as a unit circle.
Circular Functions

Definition: the sine and the cosine functions in the
set of real numbers are defined by x = cosine θ and
y = sine θ, where θ ϵ R and P(θ) = (x, y).

The x and y coordinates for each point along the circle may
be ascertained by reading off the values on the x and y axes.
If you picture a right triangle with one side along the x-axis:

Definition: the four other circular functions are
defined by the following:
tangent θ
=
cosecant θ
=
secant θ
=
cotangent θ =
𝑦
,x≠0
𝑥
1
,y≠0
𝑦
1
,x≠0
𝑥
𝑥
,y≠0
𝑦
where P(θ) = (x, y) is on the unit circle.
Notes:
 The conventional way of writing the circular
functions is sin θ, cos θ, tan θ, csc θ, sec θ,
and cot θ.
 The domain of both sine and cosine functions
is the set of real numbers R.
 The range of both sine and cosine functions is
from -1 to 1, or in notations, [-1, 1].
 The domain of tan θ and sec θ is the set for
π
which θ ≠ + nπ, n ϵ I and where x ≠ 0.
2

The domain of cot θ and csc θ is the set for
which θ ≠ nπ, n ϵ I and where y ≠ 0.
Signs of the Circular Functions
Note: r is always positive. Thus, the signs of the
values of the trigonometric functions of angle θ
are determined by the signs of x and y.

The signs of the trigonometric functions depend
upon the quadrant in which the terminal side of θ
lies.
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Some Fundamental Circular Functions Identities
Any terminal point P(θ) = (x, y) on the unit
circle satisfies the equation x2 + y2 = 1, where
θ ϵ R and since x = cos θ and y = sin θ, then
cos2 θ + sin2 θ = 1.
 Note: cos2 θ = (cos θ)2 and cos2 θ ≠ cos θ2
 Quotient relations:
𝐬𝐢𝐧 𝛉
𝐜𝐨𝐬 𝛉
tan θ =
and cot θ =
𝐜𝐨𝐬 𝛉
𝐬𝐢𝐧 𝛉
 Reciprocal relations:
sin θ • csc θ = 1
cos θ • sec θ = 1
tan θ • cot θ = 1

Examples
3π
.
2
1.
Find the circular functions of θ if θ =
2.
Given θ = –π, find the circular functions
of –π.
P(θ) lies on the line segment joining the
origin and N(4, 3). Sketch and find the
circular functions of (θ + π).
3.
Special Angles
Special Angles are those angles for which the
exact values of the circular functions are easily
π π π
obtained. We shall study the angles , , ,
6 4 3
and their multiples.
 Triangles with special angle-side relationships,
introduced in geometry are useful in evaluating
π π π
the trigonometric function values of , , , or
6 4 3
30°, 45°, and 60° respectively.

𝝅
𝟒
and its odd multiples
𝝅
𝟒
and its odd multiples
Circular functions of
𝝅
𝟒
and its odd multiples
Example:
Find the circular functions of
𝝅
.
𝟒
𝝅
𝟑
and its multiples
𝝅
𝟑
and its multiples
Circular functions of
𝝅
𝟑
and its multiples
Example:
Find the circular functions of
𝝅
.
𝟑
𝝅
𝟔
and its multiples
𝝅
𝟔
and its multiples
330°
Circular functions of
𝝅
𝟔
and its multiples
Example:
Find the circular functions of
𝝅
.
𝟔
Other illustrations…
Quadrantal Angles

An angle in standard position whose terminal
side lies on an axis (either the x-axis or the yaxis) is said to be quadrantal.
Quadrantal Angles
Example:
Find the circular functions of 2π.
The Unit Circle
The Unit Circle
Reference Angles… (a recall)
Examples: Find the reference angle for
each of the following angles.
1. 460°
2. 165°
3. -40°
4. -283°
5.
6.
5π
6
11π
4
Evaluating Trigonometric Functions
Using Reference Angles
Procedures:
1. Determine the reference angle θ’ associated
with θ.
2. Find the value of the corresponding
trigonometric function of θ’. Exact value can be
obtained if θ’ is 30°, 45°, or 60°, or it can be an
approximate value from a calculator or a table.
3. Affix the proper algebraic sign for the particular
function by determining the signs of P(x, y)
where the terminal side of θ lies.
Examples
Use reference angles to solve the following.
1. Find tan 120°.
2. Approximate the value of cos (-16°48’) by
first expressing it in terms of a function of
the associated reference angle.
3. Approximate the value of sin 220°.
Exercises
Evaluate the following using reference angles.
cos 210°
2. tan (-45°)
1.
−π
3. sec (
)
3
7π
4. csc (
)
6
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