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

Trigonometric functions

Step one: similar triangles Two similar triangles have the same set of angles, and have the properties that c a B = a A B C = b c , and A C = a θ b , c . b C A θ B

Step one: similar triangles Two similar triangles have the same set of angles, and have the properties that c a B = a A C = b B c , and A C = a θ b , c . b Define C cos( θ ) = b sin( θ ) = a A and c . θ c B

Step one: similar triangles Two similar triangles have the same set of angles, and have the properties that c a B = a A C = b B c , and A C = a θ b , c . b Define C cos( θ ) = b sin( θ ) = a A and c . θ c B Then let tan( θ ) = sin( θ ) cos( θ ) = a sin( θ ) = c 1 b , sec( θ ) = a , cos( θ ) = c 1 tan( θ ) = b 1 csc( θ ) = b , cot( θ ) = a .

Easy angles: isosceles right triangle: equilateral triangle cut in half: 1 √ 2 1 π /6 1 h π /3 π /4 1/2 1 √ 1 − (1 / 2) 2 = p h = 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: 1 √ 2 1 π /6 1 h π /3 π /4 1/2 1 √ 1 − (1 / 2) 2 = p h = 3 / 2 cos( θ ) sin( θ ) tan( θ ) sec( θ ) csc( θ ) cot( θ ) √ √ 1 1 π / 4 1 2 2 1 √ √ 2 2 √ √ 1 3 2 1 π / 3 3 2 √ √ 2 2 3 3 √ √ 3 1 1 2 π / 6 2 3 √ √ 2 2 3 3

Step two: the unit circle 1 For 0 < θ < π 2 ... (x , y) θ -1 1 -1

Step two: the unit circle 1 For 0 < θ < π 2 ... (x , y) 1 y θ x -1 1 -1

Step two: the unit circle 1 For 0 < θ < π 2 ... (x , y) 1 cos( θ ) = x 1 = x y θ sin( θ ) = y x -1 1 1 = y -1 Use this idea to extend trig functions to any θ ...

Define cos( θ ) = x sin( θ ) = y , there θ is defined by... 0 ≤ θ ≤ 2 π all θ ≥ 0 θ < 0 1 1 1 (x , y) θ θ -1 1 -1 1 -1 1 θ (x , y) (x , y) -1 -1 -1

Define cos( θ ) = x sin( θ ) = y , there θ is defined by... 0 ≤ θ ≤ 2 π all θ ≥ 0 θ < 0 1 1 1 (x , y) θ θ -1 1 -1 1 -1 1 θ (x , y) (x , y) -1 -1 -1 Sidebar: In calculus, radians are king. Where do they come from? Circumference of a unit circle: 2 π Arclength of a wedge with angle θ : θ θ 360 � ∗ 2 π (if in degrees) or 2 π ∗ 2 π = θ (if in radians)

Reading o ff of the unit circle 1 -1 1 -1 3 π 0 π π π π π 2 2 6 4 3 cos( θ ) sin( θ ) 2 π 3 π 5 π 7 π 5 π 4 π 5 π 7 π 11 π 3 4 6 6 4 3 3 4 6 cos( θ ) sin( θ )

Reading o ff of the unit circle π /2 1 π 0 -1 1 -1 3 π /2 3 π 0 π π π π π 2 2 6 4 3 cos( θ ) 1 0 − 1 0 sin( θ ) 0 1 0 − 1 2 π 3 π 5 π 7 π 5 π 4 π 5 π 7 π 11 π 3 4 6 6 4 3 3 4 6 cos( θ ) sin( θ )

Reading o ff of the unit circle π /2 1 π /3 π /4 π /6 π 0 -1 1 -1 3 π /2 3 π 0 π π π π π 2 2 6 4 3 √ 3 1 1 cos( θ ) 1 0 − 1 0 2 √ 2 2 √ 1 1 3 sin( θ ) 0 1 0 − 1 2 √ 2 2 2 π 3 π 5 π 7 π 5 π 4 π 5 π 7 π 11 π 3 4 6 6 4 3 3 4 6 cos( θ ) sin( θ )

Reading o ff of the unit circle cos( π − θ ) = − cos( θ ) sin( π − θ ) = sin( θ ) π /2 1 π /3 π /4 π /6 π 0 θ θ -1 1 -1 3 π /2 3 π 0 π π π π π 2 2 6 4 3 √ 3 1 1 cos( θ ) 1 0 − 1 0 2 √ 2 2 √ 1 1 3 sin( θ ) 0 1 0 − 1 2 √ 2 2 2 π 3 π 5 π 7 π 5 π 4 π 5 π 7 π 11 π 3 4 6 6 4 3 3 4 6 cos( θ ) sin( θ )

Reading o ff of the unit circle cos( π − θ ) = − cos( θ ) sin( π − θ ) = sin( θ ) π /2 1 2 π /3 π /3 3 π /4 π /4 5 π /6 π /6 π 0 θ θ -1 1 -1 3 π /2 3 π 0 π π π π π 2 2 6 4 3 √ 3 1 1 cos( θ ) 1 0 − 1 0 2 √ 2 2 √ 1 1 3 sin( θ ) 0 1 0 − 1 2 √ 2 2 2 π 3 π 5 π 7 π 5 π 4 π 5 π 7 π 11 π 3 4 6 6 4 3 3 4 6 √ - 1 - 1 3 cos( θ ) - 2 √ 2 2 √ 3 1 1 sin( θ ) 2 √ 2 2

Reading o ff of the unit circle cos( π − θ ) = − cos( θ ) sin( π − θ ) = sin( θ ) π /2 1 2 π /3 π /3 3 π /4 π /4 5 π /6 π /6 cos( − θ ) = cos( θ ) sin( − θ ) = − sin( θ ) θ π 0 -1 - θ 1 cos(2 π n + θ ) = cos( θ ) sin(2 π n + θ ) = sin( θ ) -1 3 π /2 3 π 0 π π π π π 2 2 6 4 3 √ 3 1 1 cos( θ ) 1 0 − 1 0 2 √ 2 2 √ 1 1 3 sin( θ ) 0 1 0 − 1 2 √ 2 2 2 π 3 π 5 π 7 π 5 π 4 π 5 π 7 π 11 π 3 4 6 6 4 3 3 4 6 √ - 1 - 1 3 cos( θ ) - 2 √ 2 2 √ 3 1 1 sin( θ ) 2 √ 2 2

Reading o ff of the unit circle cos( π − θ ) = − cos( θ ) sin( π − θ ) = sin( θ ) π /2 1 2 π /3 π /3 3 π /4 π /4 5 π /6 π /6 cos( − θ ) = cos( θ ) sin( − θ ) = − sin( θ ) θ π 0 -1 - θ 1 cos(2 π n + θ ) = cos( θ ) sin(2 π n + θ ) = sin( θ ) 7 π /6 7 π /4 11 π /6 4 π /3 5 π /4 -1 5 π /3 3 π /2 3 π 0 π π π π π 2 2 6 4 3 √ 3 1 1 cos( θ ) 1 0 − 1 0 2 √ 2 2 √ 1 1 3 sin( θ ) 0 1 0 − 1 2 √ 2 2 2 π 3 π 5 π 7 π 5 π 4 π 5 π 7 π 11 π 3 4 6 6 4 3 3 4 6 √ √ √ - 1 - 1 3 3 - 1 - 1 1 1 3 cos( θ ) - - 2 √ 2 2 √ 2 2 √ 2 2 2 2 √ √ √ 3 1 1 - 1 - 1 3 3 - 1 - 1 sin( θ ) - - 2 √ 2 2 √ 2 2 √ 2 2 2 2

Reading o ff of the unit circle cos( π − θ ) = − cos( θ ) sin( π − θ ) = sin( θ ) π /2 1 2 π /3 π /3 3 π /4 π /4 5 π /6 π /6 cos( − θ ) = cos( θ ) sin( − θ ) = − sin( θ ) π 0 -1 1 cos(2 π n + θ ) = cos( θ ) sin(2 π n + θ ) = sin( θ ) 7 π /6 7 π /4 11 π /6 4 π /3 5 π /4 -1 5 π /3 x 2 + y 2 = 1 = ⇒ cos 2 ( θ )+sin 2 ( θ ) = 1 3 π /2 3 π 0 π π π π π 2 2 6 4 3 √ 3 1 1 cos( θ ) 1 0 − 1 0 2 √ 2 2 √ 1 1 3 sin( θ ) 0 1 0 − 1 2 √ 2 2 2 π 3 π 5 π 7 π 5 π 4 π 5 π 7 π 11 π 3 4 6 6 4 3 3 4 6 √ √ √ - 1 - 1 3 3 - 1 - 1 1 1 3 cos( θ ) - - 2 √ 2 2 √ 2 2 √ 2 2 2 2 √ √ √ 3 1 1 - 1 - 1 3 3 - 1 - 1 sin( θ ) - - 2 √ 2 2 √ 2 2 √ 2 2 2 2

Plotting on the θ - y axis Graph of y = cos( θ ): 1 π - π 2 π -2 π -1 Graph of y = sin( θ ): 1 π - π 2 π -2 π -1

Plotting on the θ - y axis Graph of y = cos( θ ): 1 π - π 2 π -2 π -1 Graph of y = sin( θ ): 1 π - π 2 π -2 π -1

Plotting on the θ - y axis Graph of y = cos( θ ): 1 A π - π 2 π -2 π -1 A = Amplitude = 1 2 length of the range = 1 Graph of y = sin( θ ): 1 A π - π 2 π -2 π -1 A =Amplitude = 1 2 length of the range = 1

Plotting on the θ - y axis Graph of y = cos( θ ): 1 π - π 2 π -2 π -1 T A = Amplitude = 1 2 length of the range = 1 T =Period = time to repeat = 2 π Graph of y = sin( θ ): 1 - π π -2 π 2 π -1 T A =Amplitude = 1 2 length of the range = 1 T =Period = time to repeat = 2 π

Trig identities to know and love: Even/odd: cos( − θ ) = cos( θ ) (even) sin( − θ ) = − sin( θ ) (odd) Pythagorean identity: cos 2 ( θ ) + sin 2 ( 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(2 θ ) = cos 2 ( θ ) − sin 2 ( θ ) and sin(2 θ ) = 2 sin( θ ) cos( θ ) )

Other trig functions y = cos( θ ) y = sin( θ ) 1 1 - π π - π π -1 -1 sec( θ ) = 1 / cos( θ ) csc( θ ) = 1 / sin( θ ) 2 2 1 1 - π π - π π -1 -1 -2 -2

Other trig functions y = cos( θ ) y = sin( θ ) 1 1 - π π - π π -1 -1 tan( θ ) = sin( θ ) / cos( θ ) cot( θ ) = cos( θ ) / sin( θ ) 2 2 1 1 - π π - π π -1 -1 -2 -2