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.