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The Values of sin  , cos  , tan . Quadrants and angles in the unit circle. y. 90 °. Quadrant II. Quadrant I. 0 °. 180 °. x. 360 °. Quadrant IV. Quadrant III. 270 °. The Values of sin  , cos  , tan . Cartesian plane can be divided into four parts called quadrants .

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slide2

Quadrants and angles in the unit circle

y

90°

Quadrant II

Quadrant I

180°

x

360°

Quadrant IV

Quadrant III

270°

The Values of sin , cos, tan 

  • Cartesian plane can be divided into four parts called quadrants.
  • Quadrants are named in the anticlockwise direction.
slide3

P

The Values of sin , cos, tan 

Quadrants and angles in the unit circle

y

  • Angle  is measured by rotating the line OP in the anticlockwise direction from the positive x-axis at the origin, O.

x

O

slide4

Verify sin  = y-coordinate in quadrant I of the unit circle

P

(x, y)

y

x

The Values of sin , cos, tan 

y

sin  =

=

= y

1

x

Q

O

 sin  = y-coordinate

slide5

Verify cos  = x-coordinate in quadrant I of the unit circle

The Values of sin , cos, tan 

y

cos  =

=

= x

P

(x, y)

1

y

x

x

Q

O

 cos = x-coordinate

slide6

 tan =

The Values of sin , cos, tan 

Verify tan  = in quadrant I of the unit circle

y

tan  =

=

P

(x, y)

1

y

x

x

Q

O

slide7

Determine whether the value is positive or negative

y

x

The Values of sin , cos, tan 

  • Quadrant I = All positive

sin

All

  • Quadrant II = sin  positive

II

I

IV

III

  • Quadrant III = tan  positive

tan

cos

  • Quadrant IV = cos  positive
slide8

Example 1: sin 213°

  • Sin  is positive in quadrant II.

Not

quadrant II

The Values of sin , cos, tan 

Determine whether the value is positive or negative

y

  • The angle 213° lies in quadrant III.

213°

x

O

  • Therefore, the value of sin 213° is negative.
slide9

Example 2: cos 321°

  • Cos  is positive in quadrant IV.

It is

quadrant IV.

The Values of sin , cos, tan 

Determine whether the value is positive or negative

y

  • The angle 321° lies in quadrant IV.

321°

x

O

  • Therefore, the value of cos 321° is positive.
slide10

Example 3: tan 123°

  • Tan  is positive in quadrant III.

Not

quadrant III

The Values of sin , cos, tan 

Determine whether the value is positive or negative

y

  • The angle 123° lies in quadrant II.

123°

x

O

  • Therefore, the value of tan 123° is negative.
slide11

Example 4: sin 32°

  • All positive in quadrant I.

It is

quadrant I.

The Values of sin , cos, tan 

Determine whether the value is positive or negative

y

  • The angle 32° lies in quadrant I.

32°

x

O

  • Therefore, the value of sin 32° is positive.
slide12

Determine the values of sine, cosine and tangent for special angles

  • sin 45° =

45°

  • cos 45° =

1

45°

1

The Values of sin , cos, tan 

  • tan 45° = 1
slide13

Determine the values of sine, cosine and tangent for special angles

  • sin 30° =

30°

  • cos 30° =

2

  • tan 30° =

60°

1

The Values of sin , cos, tan 

slide14

Determine the values of sine, cosine and tangent for special angles

  • sin 60° =

30°

  • cos 60° =

2

  • tan 60° =

60°

1

The Values of sin , cos, tan 

slide15

Determine the values of sine, cosine and tangent for special angles

y

(0, 1)

x

O

(–1, 0)

(1, 0)

(0, –1)

The Values of sin , cos, tan 

slide16

Summary:

The Values of sin , cos, tan 

Determine the values of sine, cosine and tangent for special angles

slide17

Question 1: Calculate the values of the following:

7 sin 90° + 4 cos 180 °

Solution:

The Values of sin , cos, tan 

Determine the values of sine, cosine and tangent for special angles

7 sin 90° + 4 cos 180 ° =

7 × (1) + 4 × (–1)

= 7 – 4

= 3

slide18

Values of angles in quadrant II

y

P

x

O

The Values of sin , cos, tan 

 = corresponding angle in quadrant I

between x-axis and line OP

slide19

   where  = 180° – 

sin  = + sin 

cos  = – cos 

tan  = – tan 

P

 = – 90°

The Values of sin , cos, tan 

Values of angles in quadrant II

y

x

O

X

slide20

Values of angles in quadrant III

   where  =  – 180°

sin  = – sin 

cos  = – cos 

tan  = + tan 

P

 = 270°–

The Values of sin , cos, tan 

y

x

O

X

slide21

Values of angles in quadrant IV

   where  = 360° – 

P

sin  = – sin 

cos  = + cos 

tan  = – tan 

 = – 270°

The Values of sin , cos, tan 

y

x

O

X

slide22

Finding the value of an angle

Solution:

y

x

O

P

The Values of sin , cos, tan 

Question 1: Find the value of sin 231°.

  • 231°  quadrant III
  • sin 231°  negative
  • sin 231° = – sin (231° – 180°)

231°

= – sin 51°

= – 0.7771

slide23

Solution:

y

x

O

P

The Values of sin , cos, tan 

Finding the value of an angle

Question 2: Find the value of cos 303° 17‘.

  • 303° 17\'  quadrant IV
  • cos 303° 17\'  positive
  • cos 303° 17\' = cos (360° – 303° 17\')

303° 17\'

= cos 56° 43\'

= 0.5488

slide24

Solution:

y

P

x

O

The Values of sin , cos, tan 

Finding the value of an angle

Question 3: Find the value of tan 117° 13\'.

  • 117° 13\'  quadrant II
  • tan 117° 13\'  negative
  • tan 117° 13\' = – tan (180° – 117° 13\')

117° 13\'

= – tan 62° 47\'

= – 1.945

slide25

Finding angles between 0° and 360°

Solution:

P

y

y

P

x

x

72°

72°

x

x

O

The Values of sin , cos, tan 

Question 1: For sin x = 0.9511 where 0°≤x≤ 360°, find the value of x.

  • Corresponding acute angle, x = 72°
  • 0.9511  positive
  • Therefore, the acute angle is in quadrant I or II.

and

  • Quadrant I: x =

72°

180° – 72° = 108°

  • Quadrant II: x =
slide26

Finding angles between 0° and 360°

Solution:

y

y

P

x

60° 12\'

x

x

x

O

60° 12\'

P

The Values of sin , cos, tan 

Question 2: For tan x = – 1.746 where 0°≤x≤ 360°, find the value of x.

  • Corresponding acute angle, x = 60° 12\'
  • – 1.746  negative
  • Therefore, the acute angle is in quadrant II or IV.

and

  • Quadrant II: x =

180° – 60° 12\' = 119° 48\'

  • Quadrant IV: x =

360° – 60° 12\' = 299° 48\'

slide27

Solution:

y

y

P

x

60°

x

x

x

O

60°

P

The Values of sin , cos, tan 

Finding angles between 0° and 360°

Question 3: For cos x = 0.5 where 0°≤x≤ 360°, find the value of x.

  • Corresponding acute angle, x = 60°
  • 0.5  positive
  • Therefore, the acute angle is in quadrant I or IV.

and

  • Quadrant I: x =

60°

360° – 60° = 300°

  • Quadrant IV: x =
slide28

Solve problems involving sine, cosine and tangent

Question: In the diagram below, HMS and JHN are straight lines. H is the midpoint of JN. Given that HM = 12 cm, MN = 13 cm and FJ = 4 cm, calculate:

N

  • the length of HN,
  • the value of cos x°,
  • the value of tan y°.

S

H

M

F

J

The Values of sin , cos, tan 

slide29

Solution:

Pythagoras’ theorem

The Values of sin , cos, tan 

Solve problems involving sine, cosine and tangent

(a)

N

132 – 122

13 cm

HN2 =

= 169 – 144

= 25

S

H

12 cm

M

HN = 5 cm

F

J

4 cm

slide30

HMS is a straight line

The Values of sin , cos, tan 

Solve problems involving sine, cosine and tangent

Solution:

(b)

N

x° = 180° – HMN

cos x° =

– cos HMN

S

H

M

=

F

J

slide31

JHN is a straight line

The Values of sin , cos, tan 

Solve problems involving sine, cosine and tangent

Solution:

(c)

N

y° = 180° – FHJ

tan y° =

– tan FHJ

S

H

M

=

F

J

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