Young Won Lim12/12/09

● Quadrant Angle Trigonometry● Negative Angle Trigonometry● Reference Angle Trigonometry● Sinusoidal Waves

Trigonometry (3A)

Young Won Lim12/12/09

Copyright (c) 2009 Young W. Lim.

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Quadrant Trigonometry 3 Young Won Lim12/12/09

Triangle Trigonometry

a

c

b

A B

C

All Acute Angles

a

c

b

A

C

B

One Obtuse Angle

Right Triangle

ac

bA B

C

Oblique Triangle

sin = a / c

cos = b / c

tan = a / b

sin = ?

cos = ?

tan = ?

sin = ?

cos = ?

tan = ?

0 ˚ 90 ˚ 90 ˚ 180 ˚0 ˚ 90 ˚

Quadrant Trigonometry 4 Young Won Lim12/12/09

Oblique Triangles Trigonometry

asin A

=bsinB

=csinC

c2 = a2 b2−2ab cosC

a2 = b2 c2−2b c cos A

b2 = c2 a2−2c acosB

The Law of Sines

The Law of Cosines

a

c

b

A B

C

All Acute Angles

a

c

b

A

C

B

One Obtuse Angle

sin = sin 180 ˚ − = sin

cos = cos 180 ˚ − = − cos

tan = tan 180 ˚ − = − tan

Assuming

0 ˚ 90 ˚ , 0 ˚ 180 ˚

Quadrant Trigonometry 5 Young Won Lim12/12/09

Trigonometry in Quadrant Angles (1)

a

c

b

A B

C

c

(-x, y)

-x

y

sin = sin 180 ˚ − = sin

cos = cos 180 ˚ − = − cos

tan = tan 180 ˚ − = − tan

(+x, + y)

x

y

b

− cos = x / b

− tan = y / x

sin = y / b

Quadrant Trigonometry 6 Young Won Lim12/12/09

Trigonometry in Quadrant Angles (2)

a

c

b

A B

C

c

(-x, y)

-x

y

sin = sin 180 ˚ − = sin

cos = cos 180 ˚ − = − cos

tan = tan 180 ˚ − = − tan

a

c

b

A B

C

c

(-x, y)

-x

y

sin = y / b

cos = −x / b

tan = − y / x

Quadrant Trigonometry 7 Young Won Lim12/12/09

Trigonometry in Quadrant Angles (3)

(-x, y)

-x

y

a

c

b

A B

C

c

(-x, y)

-x-x

y

+r

r

r

sin = y / r

cos = −x / r

tan = − y / x

r = x2 y2

Isosceles Triangle

sin = y / b

cos = −x / b

tan = − y / x

Quadrant Trigonometry 8 Young Won Lim12/12/09

Trigonometry in Quadrant Angles (4)

(-x, y)

-x

y

+r

sin = y

cos = −x

tan = − y / x

1 = x2 y2

y

−x

−x , y

Unit Circle

r

r

−1 1

sin = y / r

cos = −x / r

tan = − y / x

r = x2 y2

Quadrant Trigonometry 9 Young Won Lim12/12/09

Negative Angle Trigonometry (1)

y

x 1

x , y

−1

1st Quadrant Angle

sin = y

cos = x

tan = y / x

−

−y

x

1

x , − y

−1

4th Quadrant Angle

sin − = − sin = −y

cos − = cos = x

tan− = − tan = − y /x

0 ̊ < α < 90 ̊ –90 ̊ < –α < 0 ̊

Quadrant Trigonometry 10 Young Won Lim12/12/09

Negative Angle Trigonometry (2)

−

−y

−x

1

−x , − y

−1

3rd Quadrant Angle

sin − = − sin = − y

cos − = − cos = −x

tan − = tan = y /x

sin = sin = y

cos = − cos = −x

tan = − tan = − y / x

y

−x 1

−x , y

−1

2nd Quadrant Angle

90 ̊ < β < 180 ̊ –180 ̊ < –β < –90 ̊

Quadrant Trigonometry 11 Young Won Lim12/12/09

Reference Angle (1)

y

−x 1

−x , y

−1

2nd Quadrant Angle θ

y

x 1

x , y

−1

1st Quadrant Angle θ

sin = y

cos = x

tan = y / x

Reference Angle α

sin = sin = y

cos = − cos = −x

tan = − tan = − y / x

= 180 ˚ −

90 ˚ 180 ˚

Quadrant Trigonometry 12 Young Won Lim12/12/09

Reference Angle (2)

−y

1

−x , − y

−1

3rd Quadrant Angle θ

−y

1

x , − y

−1

4th Quadrant Angle θ

Reference Angle α Reference Angle α

sin = − sin = − y

cos = − cos = −x

tan = tan = y / x

sin = − sin = − y

cos = cos = x

tan = − tan = − y / x

= − 180 ˚ = 360 ˚ −

−x x

270 ˚ 360 ˚180 ˚ 270 ˚

Quadrant Trigonometry 13 Young Won Lim12/12/09

Reference Angle (3)

sin = − sin

cos = − cos

tan = tan

sin = − sin

cos = cos

tan = − tan

sin = sin

cos = cos

tan = tan

sin = sin

cos = − cos

tan = − tan

Reference Angle α

All +only sin +

only tan +

only cos +

= − = 2−

= − =

A Quadrant Angle θ

Quadrant Trigonometry 14 Young Won Lim12/12/09

Making a Helix

Transparent OHP Film

Quadrant Trigonometry 15 Young Won Lim12/12/09

y

z

y

z

x

x

z

y

x

Front View

Top View

Side View

A Helix and Viewpoints

Quadrant Trigonometry 16 Young Won Lim12/12/09

y

z

y

z

y

x

Front View

Side View

0

2

32

Sine Wave

Sine Wave

Quadrant Trigonometry 17 Young Won Lim12/12/09

y

z

y

x

Top View

Side View

0

2

32

y

z

Cosine Wave

Cosine Wave

Quadrant Trigonometry 18 Young Won Lim12/12/09

Symmetry in Sinusoid

Quadrant Trigonometry 19 Young Won Lim12/12/09

Sine and Cosine Waves

2− 312

32

52−

12 0

Sine Wave

Cosine Wave

Quadrant Trigonometry 20 Young Won Lim12/12/09

Sine Wave Symmetry

2− 3

12

32

52−

12

0

Sine Wave

Quadrant Trigonometry 21 Young Won Lim12/12/09

Cosine Wave Symmetry

2− 3

12

32

52−

12

0

Cosine Wave

Young Won Lim12/12/09

References

[1] http://en.wikipedia.org/[2] http://planetmath.org/[3] Blitzer, R. “Algebra & Trigonometry.” 3rd ed, Prentice Hall[4] Smith, R. T., Minton, R. B. “Calculus: Concepts & Connections,” Mc Graw Hill [5] 홍성대, “기본/실력 수학의 정석,”성지출판