Full Transcript

Fungsi Trigonometri dengan menggunakan lingkaran satuan Matematika SMA Kelas X

10:02495 summary words · ~2 min readEnglishBy Bali MelajahTranscribed Jul 19, 2026

Summary

The video explains how to determine trigonometric function values using the unit circle, emphasizing quadrant-based sign rules for sine, cosine, and tangent.

Understanding trigonometric functions via the unit circle is essential for solving advanced math problems in physics, engineering, and computer science.

Section summaries

Intro / Definitions

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The instructor introduces the video's purpose: teaching trigonometric functions using the unit circle. They contrast algebraic functions with trigonometric ones, explaining that angles map to values via ratios. The unit circle's definition (radius 1, centered at origin) is introduced, along with quadrant divisions.

Trigonometric functions pair angles to values using ratios.
Unit circle has radius 1 and equation x² + y² = 1.
Quadrants are defined by positive/negative axes.

Establishes foundational concepts for the entire lesson.

Quadrant 1 Analysis

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The instructor demonstrates how to derive trigonometric ratios in quadrant 1 using a right triangle formed by a point on the circle. They show that sin(θ) = y, cos(θ) = x, and tan(θ) = y/x, emphasizing all values are positive here.

Quadrant 1: All trigonometric ratios positive.
Right triangle sides correspond to x, y, and hypotenuse 1.
Ratios derived directly from coordinates.

Introduces core geometric relationships for trigonometric functions.

Quadrant 2 Analysis

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The instructor explains quadrant 2 using a point with negative x and positive y. They show that sin(θ) remains positive (y-axis positive), while cos(θ) and tan(θ) become negative due to the x-axis's negativity.

Quadrant 2: Sine positive, cosine/tangent negative.
Right triangle sides retain x (negative) and y (positive).
Sign rules simplify quadrant analysis.

Critical for understanding sign changes in non-acute angles.

Quadrant 3/4 Analysis

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The instructor extends the analysis to quadrants 3 and 4. In quadrant 3, both x and y are negative, making sine and cosine negative but tangent positive. In quadrant 4, x is positive and y negative, resulting in negative sine and positive cosine/tangent.

Quadrant 3: Sine/cosine negative, tangent positive.
Quadrant 4: Sine negative, cosine/tangent positive.
Sign rules apply consistently across all quadrants.

Completes the quadrant-based sign rule framework.

Summary & Practice

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The instructor summarizes quadrant sign rules and provides a practice problem: determining signs for a point (-√3, 1) in quadrant 2. They confirm sin(θ) is positive, cos(θ) negative, and tan(θ) negative, reinforcing the rules.

Practice problems solidify quadrant sign understanding.
Real-world examples (e.g., signal strength) contextualize trigonometry.
Final summary reinforces key takeaways.

Reinforces concepts through application and summary.

Key points

Unit Circle Definition — Trigonometric functions are defined using coordinates (x, y) on a unit circle, where sin(θ) = y, cos(θ) = x, and tan(θ) = y/x.
Quadrant Sign Rules — Sine is positive in quadrants 1/2, negative in 3/4; cosine is positive in 1/4, negative in 2/3; tangent is positive in 1/3, negative in 2/4.
Right Triangle Construction — For any angle, a right triangle is formed by dropping a perpendicular from the point on the circle to the x-axis, using its sides to derive ratios.

“Sinus Alpha dalam kuadran satu positif karena sumbu y positif.” — Instructor

“Cosinus Alpha dalam kuadran dua negatif karena sumbu x negatif.” — Instructor

AI-generated from the transcript. May contain errors.

0:00

Hi honestly Hello little brothers and sisters how are

0:17

you today hopefully you are in good

0:19

health yes introduce your name

0:22

okay brother teaching mathematics at SMA

0:26

Negeri 1 Mengwi in this video we

0:29

will learn about

0:31

trigonometric functions using the

0:32

unit circle little brothers and sisters are certainly

0:36

familiar with the term

0:37

online shopping Not every one of us must have

0:41

done it Have you ever

0:45

felt irritated because the signal at home

0:48

does not support making

0:49

fast transactions everything suddenly

0:53

feels very slow until you

0:57

try to shake your cellphone

1:00

because it is to the left but still there is no

1:03

change in signal Did you know that

1:07

actually the signal is a superposition

1:10

of infinite many

1:12

trigonometric functions by analyzing

1:16

these functions separately the signal can be

1:18

better understood the

1:22

basic competency that we will discuss this time

1:24

is trigonometric functions

1:26

using the unit circle After

1:29

studying this material you are expected to be able to

1:31

determine the

1:34

comparison value of trigonometric functions on the unit

1:36

circle little brothers and sisters certainly still

1:40

remember about the function not the

1:42

algebraic function that we learned so far

1:44

pairing the set of real numbers to the

1:47

Riel number for example fx = x + 1

1:51

pairing x = 0 to f0 which has a value of

1:56

one then what is the difference with

2:00

trigonometric functions Trigonometry is a

2:03

function that pairs a set of angles

2:06

to a value using

2:08

trigonometric ratios such as sine

2:12

cosine tangent second cosecant and

2:16

cotangent example FX = Sin X pairs

2:21

angle X = 0° to sinol which is zero

2:27

for angles that are less

2:30

than or equal to 90° we can

2:33

use the concept of

2:34

trigonometric ratios in right triangles

2:37

but what about obtuse angles

2:40

that are more than 90 degrees Well

2:45

for that we will learn the concept of

2:47

trigonometric ratios by

2:49

using circles set area of a

2:51

unit circle is a circle with a radius of

2:54

one unit and centered at the origin

2:56

with the equation x squared + y

3:00

1 = 1221 previously only up to an angle of

3:05

90° which means only a quarter of a

3:08

circle Now we will start

3:11

learning for angles up to one

3:14

full rotation note the area from 0° to

3:18

90° is called quadrant 1 this area

3:22

is limited by the positive x-axis and the

3:25

positive y-axis next the area from 90°

3:30

to 180° is called quadrant 2

3:33

this area is limited by the negative X-axis

3:36

and the positive y-axis the area from 180°

3:41

to 270° is called quadrant 3 which is

3:46

limited by the negative X temperature and the negative y axis

3:49

the area from 270° to 360° is

3:54

called quadrant 4 this area is limited

3:58

by the positive x axis

4:00

and the negative y axis How

4:03

to determine the value of the

4:04

trigonometric ratio of the angles in

4:07

this quadrant let's observe together

4:11

for example point a is a point with

4:14

coordinates 1.0 which is located on the

4:16

unit circle then we can

4:20

determine any point p such

4:23

that the angle Alfa is a

4:25

positive angle from point p we can draw a

4:30

perpendicular line oia so we

4:32

get a right triangle with

4:35

the length of the side next to the angle Alfa is

4:38

x according to the abscissa of point P and the

4:42

length of the side in front of the angle Alfa is

4:44

Yes according to the ordinate of point p

4:48

and the hypotenuse is one unit long

4:51

according to the radius of the circle

4:54

through this observation we get

4:56

that the value of Sin Alfa in the quadrant

5:00

is y per unit which = y the value of cos Alfa

5:05

in the first quadrant is X1 which = X

5:10

the value of Tan Alfa in the first quadrant is y

5:15

per X the whole

5:18

trigonometric ratio is positive Because The

5:20

x-axis and y-axis in quadrant 1 are

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positive, for example Alpha is

5:26

any angle in quadrant 2, then we

5:30

can determine point p with

5:32

coordinates x comma y so that

5:35

we obtain a right triangle, the length of the

5:40

side next to angle Alpha is first, the

5:43

length of the side opposite angle Alpha is

5:45

this remains, the length of the hypotenuse is

5:48

one according to the radius of the

5:50

unit circle, we review the

5:55

trigonometric ratios that we obtained

5:56

previously, then in quadrant 2 Sin

6:00

Hi friends, positive value Because the y-axis

6:02

in family two is positive, cos

6:05

Alpha in quadrant two is negative

6:08

because the x-axis in quadrant 2 is

6:11

negative Tan Alpha in quadrant two is

6:15

negative, for example Alpha is any angle

6:19

in quadrant 3, then we can determine

6:22

point p with coordinates x comma y

6:25

so that we obtain a

6:28

right triangle, the side next to angle

6:34

Alpha is x.xi.xii in front of angle Alpha

6:38

is y and the hypotenuse is the

6:41

radius of the circle whose length is

6:43

one unit, we review the

6:46

trigonometric ratios that we

6:48

obtained previously in quadrant three, skin

6:52

Alpha is negative because the y-axis is

6:55

negative, cos Alpha is negative because

6:59

Suede is

7:00

negative and land Alpha is positive

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suppose Alpha is an angle in quadrant 4

7:09

then we can determine point p

7:12

with coordinates x comma y so

7:14

that we obtain a right triangle

7:18

[Music] the

7:20

length of the side next to the angle Alpha is

7:23

x the length of the side in front of the angle Alpha

7:26

is y and the length of the hypotenuse

7:29

is one according to the radius of

7:32

the circle we review the

7:35

trigonometric ratios that we

7:37

obtained previously then in quadrant 3 Sin

7:41

Alpha is negative because the y-axis is

7:44

negative cos Alpha is positive Because the

7:48

x-axis is positive stand Alpha is

7:52

negative because negative is positive =

7:56

negative to test the understanding of addicts so

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far let's try to do the

8:02

following question how are you guys

8:35

sure with the answer Let's

8:38

share point a with coordinates

8:43

negative not comma four located in

8:45

quadrant 2 then like the results of

8:50

our previous observations we can show

8:52

that Sin Theta is positive cos Theta is

8:56

negative and auntie is not

8:59

negative

9:00

Hi kids that was

9:03

our lesson on determining the value of

9:05

trigonometric functions on the unit circle

9:08

based on the results of our discussion earlier it

9:10

can be concluded that the

9:13

trigonometric ratio of angles located in

9:15

quadrant 1 for sine is positive

9:18

cosine is positive and tangent is also

9:21

positive

9:24

trigonometric ratio for angles located

9:26

in quadrant 2 sine value is

9:28

positive while cosine and tangent

9:30

are negative

9:33

trigonometric ratio for angles located

9:35

in quadrant 3 sine is negative

9:38

cosine is negative while tangent is

9:41

positive for

9:44

trigonometric ratio of angles located in

9:45

quadrant 4 sine is negative cosine is

9:49

positive and tangent is

9:51

negative keep up the spirit see you and

9:55

take care of your health

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