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Circular Functions (Trigonometry)Chapter 8.2:
SSMTH1: Precalculus
Science and Technology, Engineering and Mathematics (STEM) Strands
Mr. Migo M. Mendoza
Circular FunctionsLecture 8.4:
SSMTH1: Precalculus
Science and Technology, Engineering and Mathematics (STEM) Strands
Mr. Migo M. Mendoza
A Short Recap…In our lesson in Precalculus, what do you call to the circle of radius 1 and the center is
at the origin?
Unit CircleThe unit circle in the
plane is the circle of radius 1 and center at the
origin.
A Short Recap…
A unit circle is described by what
equation?
Unit Circle
122 yx
Derivation of the Six Circular Functions
Let be a point which lies on this circle. Draw the line segment
which joins P to the origin. Let θ be the measure of the angle formed
by this segment with the positive ray of the x-axis.
OP
yxP ,
Unit Circle
Trigonometric PointA trigonometric point
is a point on the unit circle. It is the intersection of the
terminal side of an angle in standard position and the unit
circle.
Pyx ),(
Take Note:For an angle θ in standard position, define the sine and the cosine functions
of the angle θ as follows:
ysin xcos
Did you know?Since the coordinate x and y of the
point P(θ) are unique for an angle θ in standard position, the above equations are actually defined as
functions.
A Short Recap…What are the definitions of
sine and cosine given by your Math teacher when you
were in Grade 9?
Sine
hypotenuse
deoppositesisin
Cosine
hypotenuse
deadjacentsicos
Tangent
deadjacentsi
deoppositesitan
Cosecant
deoppositesi
hypotenusecsc
Secant
deadjacentsi
hypotenusesec
Cotangent
deoppositesi
deadjacentsicot
Take Note: Since the coordinates of the point P depend on the
measure of the angle θ, we usually denote this point by P(θ). We will also sometimes use the
phrase “the angle θ lies in Quadrant _______” to mean that the point P(θ) which
lies on the terminal side of the angle is in the indicated quadrant.
Example 8.4.1:
Find:
6
7P
Example 8.4.1 (Graph):
The 30°-60°-90° Triangle Theorem
In a 30°-60°-90°triangle the sides are in
the ratio
.3:2:1
Take Note:
The opposite side to 30° is half the
measurement of the hypotenuse.
Take Note:
The opposite side to 60° is the measurement of the
opposite side to 30°multiply by .3
Final Answer:
Hence ,
2
1,
2
3
6
7P
Pyx ),(
Example 8.4.2: Find the values of the six
circular functions of the angle
45
Example 8.4.2 (Graph):
The Converse Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite them are
also congruent.
Take Note:Since the other acute angle also measures 45°, so that we have an
isosceles right triangle, and x = y. Hence, since P(45°) = (x, y) lies on the unit circle, we can use the equation:
45
122 yx
Final Answer:We have:
145tan
45sin
2
245cos
2
2
245csc
245sec
145cot
Example 8.4.3: Let θ be an angle in standard
position whose terminal side contains the point .Find
the values of the circular functions of θ.
)8,6(A
Example 8.4.3 (Graph):
Similar Triangle We can define similar triangles as
triangles that have three (3) pairs of congruent angles and three (3) pairs of
proportional sides.
,
,
,
FC
EB
DA
DF
AC
EF
BC
DE
AB
Final Answer:We have:
3
4tan
5
4sin
5
3cos
4
5csc
3
5sec
4
3cot
Example 8.4.4:
If and find the values of the other circular functions of θ.
13
5sin ,0cot
Take Note: Since it follows that P(θ) is either
in the third or in the fourth quadrant. Since
it follows that x and y have the same sign, and hence are both negative. Therefore, P(θ)
is in the third quadrant, where both the tangent and the cotangent functions have positive values, and the four remaining circular functions have
negative values.
,013
5sin y
,0cot y
x
Final Answer:We have:
12
5tan
13
12cos
5
13csc
12
13sec
5
12cot
θ sinθ cosθ tanθ cotθ secθ cscθ
2
1 1
2
Table 1. Values of the Six Circular Functions and Special Acute Angles
302
3
3
3
2
2
3
32
2
1
2
3
2
1
45
60
2
2
2
3
3
3
3
3
32
2
Take Note:We only need to consider the quadrant which contains the
point P(θ), since this will determine the appropriate sign
of each function.
Table 2. Signs of the Circular Functions in the Four Quadrants
P(θ) sinθ cosθ tanθ cotθ secθ cscθ
Quadrant I + + + + + +
Quadrant II + ̶ ̶ ̶ ̶ +
Quadrant III ̶ ̶ + + ̶ ̶
Quadrant IV ̶ + ̶ ̶ + ̶
Example 8.4.5.Find the values of the six
circular functions of
.6
23
Example 8.4.5 (Graph):
Final Answer:We have:
3
3
6tan
2
3
6cos
2
1
6sin
26
csc
3
32
6sec
36
cot
Example 8.4.6.
If and
find the values of the other circular
functions of θ.
24
7tan
,2)( QP
Final Answer:We have:
25
24cos
25
7sin
7
25csc
24
25sec
7
24cot
A Short Recap…
When can we say an angle is a quadrantal
angle?
θ sinθ cosθ tanθ cotθ secθ cscθ
0 1 0 undefined 1 Undefined
1 0 undefined 0 undefined 1
0 -1 0 undefined -1 Undefined
-1 0 undefined 0 undefined -1
Table 3. Values of the Circular Functions of the Basic Quadrantal Angles
0
290
180
2
3270
Example 8.4.7:Evaluate the following
expressions:
540tan5180sec2
Final Answer:
2540tan5180sec2
Example 8.4.8:Evaluate the following
expressions:
180cos5270sin4360tan 2
Final Answer:
1180cos5270sin4360tan 2
Example 8.4.9:Evaluate the following
expressions:22
2
3cos
2sin
Final Answer:
12
3cos
2sin
22
Example 8.4.10:Evaluate the following
expressions:
3cos32
3csc4
Final Answer:
73cos32
3csc4
Example 8.4.11:Evaluate the following
expressions:
540cos4450sin3 34
Final Answer:
7540cos4450sin3 34
Example 8.4.12:Evaluate the following
expressions:
3cos2sin3sec 22
Final Answer:
03cos2sin3sec 22
Classroom Task 8.3:
Please answer "Let's Practice (LP)"
Numbers 34 and 35.