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Coordinates of Terminal Points of 

Special Angles

and

Digital Lesson

Copyright © Ronaldo T. Rigor. 2010 All rights reserved.

The Sine and the

Cosine Functions

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Copyright © Ronaldo T. Rigor 2010 2

/4

/ 2

3/4

5/4

3/2

045° (1, 0)

(0,1)

(-1, 0)

(0, -1)

, 2

( , )

7/4

Coordinates of Terminal Points

of Special Angles

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r =1y

x45°

45°

x2 + y2 = r2

But, x = y.

Copyright © Ronaldo T. Rigor 2010

Why?

x2 + x2 = 12

2x2= 1

x2 = ±

x = ±

x = ±

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Copyright © Ronaldo T. Rigor 2010 4

/4

/ 2

)3/4

5/4

3/2

045° (1, 0)

(0,1)

(-1, 0)

(0, -1)

,2

( , )

7/4

(- ,

(- , - )(  , - )

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5

30° 0

/6

/3/2

2/3

5/6

7/6

4/33/2

5/3

11/6

Copyright © Ronaldo T. Rigor 2010

(  , )

60°

, 2

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6Copyright © Ronaldo T. Rigor 2010

60° 60°

30° 30°1 1

1

1/2 1/2

By Pythagorean Theorem:

x2 + y2 = r 2

Consider the triangle below:

60°

30°

y =

x

In a special triangle 30°-60°-90°,

O pposite the 30° angle is a side

which is equal to ½ .

What is the value of x?

x2 = r 2   y2

r =1

x2= (1)2   (½ )2

x2 =

x =

x = ±

60°1/2

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8Copyright © Ronaldo T. Rigor 2010

60° 60°

30° 30°

1 1

1

1/2 1/2

By Pythagorean Theorem:

x2 + y2 = r 2

Consider the triangle below

30°

60°

x = 1/2

y

In a special triangle 30°-60°-90°,

O pposite the 30° angle is a side

which is equal to ½ .

What is the value of y?

y2 = r 2   x2

r =

1

y2= (1)2   (½ )2

y2

=

y =

y = ±

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9

0

/6

/3/2

2/3

5/6

7/6

4/33/2

5/3

11/6

Copyright © Ronaldo T. Rigor 2010

(  , )

(  , )

(  , - )

(  , - )

(- , )

(- , )

(- , - )

(- ,- )

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Copyright © Ronaldo T. Rigor 2010

Find the coordinates of the terminal point of the ff. angles:

1. 11/4

2. -7/2

3. 17/6

4. -11/6

5. 23/6

6. -7/2

7. 31/2

8. -

9. 4/3

10. 10/3

11. ± 7/3

12. 38/4

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11Copyright © Ronaldo T. Rigor 2010

The Sine and Cosine Function

Consider the given right triangle ABC

A

B

C

In terms of angle A

side BC or a is the opposite side

side AC or b is the adjacent side

and side AB or c is the hypotenuse

ca

 b

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12Copyright © Ronaldo T. Rigor 2010

In terms of angle

y is the opposite side

x is the adjacent side

r is the hypotenuse

r =1 y

x

sin = y

cos = x

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Copyright © Ronaldo T. Rigor 2010 13

/4

/ 2

)3/4

5/4

3/2

045° (1, 0)

(0,1)

(-1, 0)

(0, -1)

,2

( , )

7/4

(- ,

(- , - )(  , - )

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14

0

/6

/3/2

2/3

5/6

7/6

4/33/2

5/3

11/6

Copyright © Ronaldo T. Rigor 2010

(  , )

(  , )

(  , - )

(  , - )

(- , )

(- , )

(- , - )

(- ,- )

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15Copyright © Ronaldo T. Rigor 2010

Find the value of the ff. Give your answer in rational/irrationasl form.

1. sin 14/6

2. cos 53/6

3. 2 sin /2 cos /4

4. 1 ± sin 7/4

1 + cos 3/2

5. sin 11/6 ± cos /6

sin (-/6) + cos 5/6

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16Copyright © Ronaldo T. Rigor 2010

QUIZ

A. Find the coordinates of the following:

1. 13/4 6. - 7/3

2. ± 9/4  7. 5/2

3. 19/6 8. - 3/2

4. ±13/6 9. 13

5. 8/3 10. - 45/4

B. Evaluate the following. Give your answer in rational/irrational

form.

1. sin 7/6 6.

2. cos 5/3

3. sin 5/4

4. cos (-3/4) 7.

5. cos -9/2