Homework: Graphs & Trig Equations 1.State the amplitude, period & then sketch the graph of (a) y = 3 cos 5x + 10 ≤ x ≤ 90 (b)y = ½ sin 2x0 ≤ x ≤ 360

Homework: Graphs & Trig Equations 1.(a) y = 3 cos 5x + 10 ≤ x ≤ 90 Amplitude, a =, Period = 360 = 5 Max = c + a Max = = 4 Min = c – a Min = 1 – 3 = o 3

Homework: Graphs & Trig Equations 1.(b) y = ½ sin 2x0 ≤ x ≤ 360 Amplitude, a =, Period = 360 = 2 Max = c + a Max = 0 + ½ = ½ Min = c – a Min = 0 – ½ = – ½ ½ -½ ½ 180 o

Homework: Graphs & Trig Equations 2(a) Amplitude = 5 – (-3) = 8 = 4 Period = 180 o 2 2 (2 graphs in 360) y =4 cos (2x) + c y = a cos (bx) + c y =4 cos (2x) + 1 Max = c + a 5 = c + 4 C = 1

Homework: Graphs & Trig Equations 2(b) Amplitude = 1 – (-1) = 2 = 1 Period = 720 o 2 2 (½ graph in 360) y =1 cos (½x) + c y = a cos (bx) + c y =cos ( ½ x) Max = c + a 1 = c + 1 C = 0

Homework: Graphs & Trig Equations 2(b) Amplitude = doesn’t exist Period = π/4 (1 rep in π/4 rather than 1 in π  b = 4) y = tan (4x) + c y = tan (bx) + c y =tan (4x) + 1 Normally starts at (0,0) so pushed up 1 position  c = 1

Homework: Graphs & Trig Equations 5(a) Sin 315(b) tan 7π / 6 (c) Cos(-150) = - Sin 45 = - Cos 30 = tan π / 6 = - 1 √2 = 1 √3 = - √3 2

Homework: Graphs & Trig Equations 6(a) cos – sin 2 30 (b) sin ( π / 3 ) sin ( 5π / 4 ) = (- Cos 30) 2 – (sin 30) 2 = sin ( π / 3 ) x -sin ( π / 4 ) = 3 – = √3 x -1 2 √2 = ( -√3 / 2 ) 2 – ( ½ ) 2 = 1 2 = -√3 2√2

Homework: Graphs & Trig Equations 6(c) 1 – 2 sin = 1 – 2 x (- Sin 60) 2 = 1 – 2 x ( 3 / 4 ) = 1 – 2 x ( -√3 / 2 ) 2 = - ½ = 1 – 3 / 2

Homework: Graphs & Trig Equations 9(a) Sin 3x = ≤ x ≤ 90 1 st Quadrant:3x = Sin -1 (0.32) 3x = nd Quadrant:3x = 180 – 18.7 = T SA C 3x = 18.7 ; x = 6.2 ; 53.8 Remember you must change the range:0 ≤ x ≤ 90 0 ≤ 3x ≤ 270

Homework: Graphs & Trig Equations 9(c) 2 + 3Cos(4t – 30) = 5 0 ≤ t ≤ 90 1 st Quad: (4t – 30) = Cos -1 (1) = 0 3Cos(4t – 30) = 3 4 th Quad: 360 – 0 = 360 T SA C (4t – 30) = 0, 360 t = 7.5 Remember the range:0 ≤ t ≤ 90 (4(0) – 30) ≤ (4t – 30) ≤ (4(90) – 30) – 30 ≤ (4t – 30) ≤ 330 Cos(4t – 30) = 1 4t = 30

Homework: Graphs & Trig Equations 9(e) 4 Sin 2 x = 1 0 ≤ x ≤ 2π 1 st Quad: x = Sin -1 ( ½ ) = π / 6 Sin 2 x = ¼ 2 nd Quad: π – π / 6 = 5π / 6 T SA C Sin x = ± ½ 3 rd Quad: π + π / 6 = 7π / 6 4 th Quad: 2π – π / 6 = 11π / 6 x = π / 6 ; 5π / 6 ; 7π / 6 ; 11π / 6 ** As both signs  in all 4 quadrants

Homework: Graphs & Trig Equations 9(g) 2Cos 2 t – 3Cos t + 1 = 0 0 ≤ t ≤ st Quad: t = Cos -1 ( ½ ) = 60 1 st Quad: t = Cos -1 ( 1 ) = 0 (2Cost – 1)(Cost – 1) = 0 T SA C 2Cost – 1 = 0 Cost – 1 = 0 4 th Quad: 360 – 60 = th Quad: 360 – 0 = 360 T SA C t = 0 ; 60 ; 300 ; 360 Cost = ½ Cost = 1