Trigonometric functions Step one: similar triangles Two similar - - PowerPoint PPT Presentation

Trigonometric functions

Step one: similar triangles

θ a b c θ C B A Two similar triangles have the same set

• f angles, and have the properties that

A B = a b, B C = b c , and A C = a c .

Step one: similar triangles

θ a b c θ C B A Two similar triangles have the same set

• f angles, and have the properties that

A B = a b, B C = b c , and A C = a c . Define cos(θ) = b c and sin(θ) = a c .

Step one: similar triangles

θ a b c θ C B A Two similar triangles have the same set

• f angles, and have the properties that

A B = a b, B C = b c , and A C = a c . Define cos(θ) = b c and sin(θ) = a c . Then let tan(θ) = sin(θ) cos(θ) = a b, sec(θ) = 1 sin(θ) = c a, csc(θ) = 1 cos(θ) = c b, cot(θ) = 1 tan(θ) = b a.

Easy angles:

isosceles right triangle: equilateral triangle cut in half:

√2 1 1 π/4

1 1/2 1

π/3 π/6

h h = p 1 − (1/2)2 = √ 3/2

cos(θ) sin(θ) tan(θ) sec(θ) csc(θ) cot(θ) π/4 π/3 π/6

Easy angles:

isosceles right triangle: equilateral triangle cut in half:

√2 1 1 π/4

1 1/2 1

π/3 π/6

h h = p 1 − (1/2)2 = √ 3/2

cos(θ) sin(θ) tan(θ) sec(θ) csc(θ) cot(θ) π/4 1 √ 2 1 √ 2 1 √ 2 √ 2 1 π/3 1 2 √ 3 2 √ 3 2 2 √ 3 1 √ 3 π/6 √ 3 2 1 2 1 √ 3 2 √ 3 2 √ 3

Step two: the unit circle

• 1

1

• 1

1

θ (x , y)

For 0 < θ < π

2 ...

Step two: the unit circle

• 1

1

• 1

1

θ (x , y) y x 1

For 0 < θ < π

2 ...

Step two: the unit circle

• 1

1

• 1

1

θ (x , y) y x 1

For 0 < θ < π

2 ...

cos(θ) = x 1 = x sin(θ) = y 1 = y Use this idea to extend trig functions to any θ...

Define cos(θ) = x sin(θ) = y, there θ is defined by... 0 ≤ θ ≤ 2π all θ ≥ 0 θ < 0

• 1

1

• 1

1

θ (x , y)

• 1

1

• 1

1

θ (x , y)

• 1

1

• 1

1

θ (x , y)

Define cos(θ) = x sin(θ) = y, there θ is defined by... 0 ≤ θ ≤ 2π all θ ≥ 0 θ < 0

• 1

1

• 1

1

θ (x , y)

• 1

1

• 1

1

θ (x , y)

• 1

1

• 1

1

θ (x , y)

Sidebar: In calculus, radians are king. Where do they come from? Circumference of a unit circle: 2π Arclength of a wedge with angle θ:

θ 360 ∗ 2π

(if in degrees)

• r

θ 2π ∗ 2π = θ

(if in radians)

Reading off of the unit circle

• 1

1

• 1

1

π 2

π

3π 2 π 6 π 4 π 3

cos(θ) sin(θ)

2π 3 3π 4 5π 6 7π 6 5π 4 4π 3 5π 3 7π 4 11π 6

cos(θ) sin(θ)

Reading off of the unit circle

• 1

1

• 1

1 π/2 π 3π/2

π 2

π

3π 2 π 6 π 4 π 3

cos(θ) 1 −1 sin(θ) 1 −1

2π 3 3π 4 5π 6 7π 6 5π 4 4π 3 5π 3 7π 4 11π 6

cos(θ) sin(θ)

Reading off of the unit circle

• 1

1

• 1

1 π/4 π/3 π/6 π/2 π 3π/2

π 2

π

3π 2 π 6 π 4 π 3

cos(θ) 1 −1

√ 3 2 1 √ 2 1 2

sin(θ) 1 −1

1 2 1 √ 2 √ 3 2 2π 3 3π 4 5π 6 7π 6 5π 4 4π 3 5π 3 7π 4 11π 6

cos(θ) sin(θ)

Reading off of the unit circle

• 1

1

• 1

1 π/4 π/3 π/6 π/2 π 3π/2

θ θ

cos(π−θ) = − cos(θ) sin(π−θ) = sin(θ)

π 2

π

3π 2 π 6 π 4 π 3

cos(θ) 1 −1

√ 3 2 1 √ 2 1 2

sin(θ) 1 −1

1 2 1 √ 2 √ 3 2 2π 3 3π 4 5π 6 7π 6 5π 4 4π 3 5π 3 7π 4 11π 6

cos(θ) sin(θ)

Reading off of the unit circle

• 1

1

• 1

1 π/4 π/3 π/6 π/2 π 3π/2

θ θ 5π/6 3π/4 2π/3

cos(π−θ) = − cos(θ) sin(π−θ) = sin(θ)

π 2

π

3π 2 π 6 π 4 π 3

cos(θ) 1 −1

√ 3 2 1 √ 2 1 2

sin(θ) 1 −1

1 2 1 √ 2 √ 3 2 2π 3 3π 4 5π 6 7π 6 5π 4 4π 3 5π 3 7π 4 11π 6

cos(θ)

• 1

2

• 1

√ 2

• √

3 2

sin(θ)

√ 3 2 1 √ 2 1 2

Reading off of the unit circle

• 1

1

• 1

1 π/4 π/3 π/6 π/2 π 3π/2

θ

• θ

5π/6 3π/4 2π/3

cos(π−θ) = − cos(θ) sin(π−θ) = sin(θ) cos(−θ) = cos(θ) sin(−θ) = − sin(θ) cos(2πn+θ) = cos(θ) sin(2πn+θ) = sin(θ)

π 2

π

3π 2 π 6 π 4 π 3

cos(θ) 1 −1

√ 3 2 1 √ 2 1 2

sin(θ) 1 −1

1 2 1 √ 2 √ 3 2 2π 3 3π 4 5π 6 7π 6 5π 4 4π 3 5π 3 7π 4 11π 6

cos(θ)

• 1

2

• 1

√ 2

• √

3 2

sin(θ)

√ 3 2 1 √ 2 1 2

Reading off of the unit circle

• 1

1

• 1

1 π/4 π/3 π/6 π/2 π 3π/2

θ

• θ

5π/6 3π/4 2π/3 7π/6 5π/4 4π/3 5π/3 11π/6 7π/4

cos(π−θ) = − cos(θ) sin(π−θ) = sin(θ) cos(−θ) = cos(θ) sin(−θ) = − sin(θ) cos(2πn+θ) = cos(θ) sin(2πn+θ) = sin(θ)

π 2

π

3π 2 π 6 π 4 π 3

cos(θ) 1 −1

√ 3 2 1 √ 2 1 2

sin(θ) 1 −1

1 2 1 √ 2 √ 3 2 2π 3 3π 4 5π 6 7π 6 5π 4 4π 3 5π 3 7π 4 11π 6

cos(θ)

• 1

2

• 1

√ 2

• √

3 2

• √

3 2

• 1

√ 2

• 1

2 1 2 1 √ 2 √ 3 2

sin(θ)

√ 3 2 1 √ 2 1 2

• 1

2

• 1

√ 2

• √

3 2

• √

3 2

• 1

√ 2

• 1

2

Reading off of the unit circle

• 1

1

• 1

1 π/4 π/3 π/6 π/2 π 3π/2 5π/6 3π/4 2π/3 7π/6 5π/4 4π/3 5π/3 11π/6 7π/4

cos(π−θ) = − cos(θ) sin(π−θ) = sin(θ) cos(−θ) = cos(θ) sin(−θ) = − sin(θ) cos(2πn+θ) = cos(θ) sin(2πn+θ) = sin(θ) x2+y2 = 1 = ⇒ cos2(θ)+sin2(θ) = 1

π 2

π

3π 2 π 6 π 4 π 3

cos(θ) 1 −1

√ 3 2 1 √ 2 1 2

sin(θ) 1 −1

1 2 1 √ 2 √ 3 2 2π 3 3π 4 5π 6 7π 6 5π 4 4π 3 5π 3 7π 4 11π 6

cos(θ)

• 1

2

• 1

√ 2

• √

3 2

• √

3 2

• 1

√ 2

• 1

2 1 2 1 √ 2 √ 3 2

sin(θ)

√ 3 2 1 √ 2 1 2

• 1

2

• 1

√ 2

• √

3 2

• √

3 2

• 1

√ 2

• 1

2

Plotting on the θ-y axis

Graph of y = cos(θ):

• 1

1

π 2π

• π
• 2π

Graph of y = sin(θ):

• 1

1

π 2π

• π
• 2π

Plotting on the θ-y axis

Graph of y = cos(θ):

• 1

1

π 2π

• π
• 2π

Graph of y = sin(θ):

• 1

1

π 2π

• π
• 2π

Plotting on the θ-y axis

Graph of y = cos(θ):

• 1

1

π 2π

• π
• 2π

A

A = Amplitude = 1

2 length of the range = 1

Graph of y = sin(θ):

• 1

1

π 2π

• π
• 2π

A

A=Amplitude = 1

2 length of the range = 1

Plotting on the θ-y axis

Graph of y = cos(θ):

• 1

1

π 2π

• π
• 2π

T

A = Amplitude = 1

2 length of the range = 1

T=Period = time to repeat = 2π Graph of y = sin(θ):

• 1

1

π 2π

• π
• 2π

T

A=Amplitude = 1

2 length of the range = 1

T=Period = time to repeat = 2π

Trig identities to know and love:

Even/odd: cos(−θ) = cos(θ) (even) sin(−θ) = − sin(θ) (odd) Pythagorean identity: cos2(θ) + sin2(θ) = 1 Angle addition: cos(θ + φ) = cos(θ) cos(φ) − sin(θ) sin(φ) sin(θ + φ) = sin(θ) cos(φ) + cos(θ) sin(φ)

(in particular cos(2θ) = cos2(θ) − sin2(θ) and sin(2θ) = 2 sin(θ) cos(θ))

Other trig functions

y = cos(θ) y = sin(θ)

• 1

1

π

• π
• 1

1

π

• π

sec(θ) = 1/ cos(θ) csc(θ) = 1/ sin(θ)

• 2
• 1

1 2

π

• π
• 2
• 1

1 2

π

• π

Other trig functions

y = cos(θ) y = sin(θ)

• 1

1

π

• π
• 1

1

π

• π

tan(θ) = sin(θ)/ cos(θ) cot(θ) = cos(θ)/ sin(θ)

• 2
• 1

1 2

π

• π
• 2
• 1

1 2

π

• π