Download Circular Functions

Circular Functions
by CHED on June 16, 2017
lesson duration of 2 minutes
under Precalculus
generated on June 16, 2017 at 05:01 am
Tags: Trigonometry
CHED.GOV.PH
K-12 Teacher's Resource Community
Generated: Jun 16,2017 01:01 PM
Circular Functions
( 2 mins )
Written By: CHED on July 3, 2016
Subjects: Precalculus
Tags: Trigonometry
Resources
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Content Standard
Key concepts of circular functions, trigonometric identities, inverse trigonometric functions, and the polar coordinate
system
Performance Standard
Formulate and solve accurately situational problems involving circular functions
Apply appropriate trigonometric identities in solving situational problems
Formulate and solve accurately situational problems involving appropriate trigonometric functions
Formulate and solve accurately situational problems involving the polar coordinate system
Learning Competencies
Convert degree measure to radian measure and vice versa
Introduction 1 mins
We define the six trigonometric function in such a way that the domain of each function is the set of angles in standard
position.
angles
measured
either
in degrees
radians.
In this
lesson,
modify
these
trigonometric
position.
TheThe
angles
areare
measured
either
in degrees
or or
radians.
In this
lesson,
wewe
willwill
modify
these
trigonometric
functions so that the domain will be real numbers rather than set of angles.
Teaching Notes
The teacher can give a review of trigonometric ratios as discussed in Grade 9.
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Circular Functions on Real Numbers 0 mins
Recall
cosine
functions
others:
tangent,
cosecant,
secant,
cotangent)
of angles
Recall
that that
the the
sinesine
and and
cosine
functions
(and(and
fourfour
others:
tangent,
cosecant,
secant,
and and
cotangent)
of angles
measuring
between
0
degrees
and
90
degrees
were
defined
in
the
last
quarter
of
Grade
9
as
ratios
of
sides
ofaaright
right
measuring between 0 degrees and 90 degrees were defined in the last quarter of Grade 9 as ratios of sides of
triangle. It can be verified that these definitions are special cases of the following definition.
Example 3.2.1. Find the values of cos 135 degrees, tan135 degrees, sin(?60 degrees), and sec(?60 degrees).
Solution. Refer to Figure 3.6(a).
From
properties
degrees-45
degrees
and
degrees-60
degrees
right
triangles
(with
hypotenuse
1 unit),
From
properties
of of
4545
degrees-45
degrees
and
3030
degrees-60
degrees
right
triangles
(with
hypotenuse
1 unit),
wewe
obtain the lengths of the legs as in Figure 3.6(b). Thus, the coordinates of A and B are
Teaching Notes
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45-degreeright
righttriangle
triangleisisisosceles.
isosceles.Moreover,
Moreover,the
theopposite
oppositeside
sideofofthe
the30
30degree-angle
degree-angleininaa30
30degree-60
degree-60degree
degree
AA45-degree
right triangle is half the length of its hypotenuse.
Therefore, we get
From the last example, we may then also say that
and so on.
From
the above
definitions,
we define
the six
same
six functions
real numbers.
functions
are called
From the
above
definitions,
we define
the same
functions
on realonnumbers.
These These
functions
are called
trigonometric functions.
functions.
From the last example, we then have
cos (pi/4) = cos (pi/4 rad) = cos 45 degrees = (square root of 2)/2
and
sin (-pi/3) = sin (-pi/3 rad) = sin (-60 degrees) = -(square root of 3)/2.
In the same way, we have
tan 0 = tan (0 rad) = tan 0 degrees = 0.
Example 3.2.2. Find the exact values of sin 3pi/2, cos 3pi/2, and tan 3pi/2 .
Solution. Let
LetP(3pi/2)
P(3pi/2)be
bethe
thepoint
pointon
onthe
theunit
unitcircle
circleand
andon
onthe
theterminal
terminalside
sideofofthe
theangle
angleininthe
thestandard
standardposition
positionwith
with
measure 3pi/2 rad. Then P(3pi/2) = (0,-1), and so
sin 3pi/2 = -1, cos 3pi/2 = 0
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but tan 3pi/2 is undefined.
Example 3.2.3. Suppose s is a real number such that sin s = ?3/4 and cos s > 0. Find cos s.
Solution. We may consider s as
as the
the angle
angle with
with measure
measure s rad.
rad. Let
Let P(s)
P(s) == ((x
x, y)) be
be the
the point
point on
on the
the unit
unit circle
circle and
and on
on
the terminal side of angle s.
Since P(s
P(s) is on the unit circle, we know that x^2 + ^y2 = 1. Since sin s = y = -3/4, we get
Since cos s = x > 0, we have cos s =square root of 7/4 .
Let P(x1, y1) and Q(x, y)) be
be points
points on
on the
the terminal
terminal side
side of
of an
an angle
angle ?? in
in standard
standard position,
position, where
where P is on the unit circle
and Q on the circle of radius r (not
(not necessarily
necessarily 1)
1) with
with center
center also
also at
at the
the origin,
origin, as
as shown
shown above.
above. Observe
Observe that
that we
we can
can
use similar triangles to obtain
cos ? = x1 = x1/1 = x/r and sin ? = y1 =y1/1=y/r.
We may then further generalize the definitions of the six circular functions.
We then have a second solution for Example 3.2.3 as follows. With sin s = ?3/4 and sin s = y/r, we may choose y = ?3
andr r==44(which
(whichisisalways
alwayspositive).
positive).InInthis
thiscase,
case,we
wecan
cansolve
solvefor
forx,x,which
whichisispositive
positivesince
sincecos
cosss==x/4
x/4isisgiven
giventotobe
be
and
positive.
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4 = square root of (x^2 + (-3)^2) => x = square root of 7 => cos s = (square root of 7)/4
Seatwork 0 mins
Seatwork/Homework
1.Given ?, find the exact values of the six circular functions.
2. Given
a value
of one
circular function and sign of another function (or the quadrant where the angle lies), find the
value
of the
indicated
function.
Reference Angle 1 mins
We observe that if ?1 and ?2 are coterminal angles, the values of the six circular or trigonometric functions at ?1 agree
with the values at ?2. Therefore, in finding the value of a circular function at a number ?, we can always reduce ? to a
number between 0 and 2pi. For example, sin (14pi)/3 = sin (14pi/3 – 4pi) = sin (2pi/3). Also, observe from Figure 3.7
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that sin 2pi/3 = sin pi/3.
In general, if ? 1, ? 2, ?3, and ?4 are as shown in Figure 3.8 with P(? 1) = (x
(x1, y1), then each of the x-coordinates of P(?
),
P(?
?
),
and
P(?
?
)
is
±x1,
while
the
y-coordinate
is
±y1.
The
correct
sign
is determined by the location of the angle.
P(
P(
2
3
4
Therefore, together with the correct sign, the value of a particular circular function at an angle ? can be determined by
its value at an angle ?1 with radian measure between 0 and pi/2. The angle ? 1 is called the reference angle of ?.
The signs of the coordinates of P(?) depends on the quadrant or axis where it terminates. It is important to know the
sign of each circular function in each quadrant. See Figure 3.9. It is not necessary to memorize the table, since the
sign of each function for each quadrant is easily determined from its definition. We note that the signs of cosecant,
secant, and cotangent are the same as sine, cosine, and tangent, respectively.
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Using the fact that the unit circle is symmetric with respect to the x-axis, the y-axis, and the origin, we can identify the
coordinates of all the points using the coordinates of corresponding points in the Quadrant I, as shown in Figure 3.10
for the special angles.
Example 3.2.4. Use reference angle and appropriate sign to find the exact value of each expression.
(1)sin (11pi/6) and cos (11pi/6)
(2)cos (-7pi/6)
(3)sin 150 degrees
(4)tan 8pi/3
Solution. (1) The reference angle of 11pi/6 is pi/6, and it lies in Quadrant IV wherein sine and cosine are negative and
positive, respectively.
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(2) The angle ?7pi/6 lies in Quadrant II wherein cosine is negative, and its reference angle is pi/6.
cos (-7pi/6) = - cos pi/6 = - (square root of 3)/2
(3) sin 150 degrees = sin 30 degrees = 1/2
(4) tan 8pi/3 = - tan pi/3 = - (sin pi/3)/(cos pi/3) = -((square root of 3)/2)/(1/2) = - square root of 3
Seatwork/Homework 3.2.2
Use reference angle and appropriate sign to find the exact value of each expression.
Exercises 3.2
1. Find the exact value.
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2. Find the exact value of each expression.
Teaching Notes
(sin x)^2 is denoted by sin^2 x. Similarly, this notation is used with the other trigonometric functions. In general, for a
positive integer n, sin^n x = (sin x)^n.
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3. Compute P(?), and find the exact values of the six circular functions.
4. Given the value of a particular circular function and an information about the angle ?, find the values of the other
circular functions.
Download Teaching Guide Book 0 mins
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