Trig Graphs And equations

Trigonometry Graphs to equations KUS objectives BAT know the key features of trig graphs using radians BAT solve trig equations using radians Starter: Simplify (without a calculator) sin 30 1+ 3 + 3 cos 60 2 sin 45 + cos 45 1+ 2 tan 150 1+ 3 − 3 tan 120

These are the Trigonometric graphs, but with radians instead… y y = sinθ 1 θ -360º -2π -270º -3π 2 -180º -π -90º -π 2 π 2 90º 180º π 3π 2 270º 360º 2π -1 y y = cosθ 1 θ -360º -2π -270º -3π 2 -180º -π -90º -π 2 π 2 90º 180º π 270º 3π 2 360º 2π -1 y = tanθ 1 θ -360º -2π -270º -3π 2 -180º -π -90º -π 2 π 2 90º 180º π 3π 2 270º 360º 2π -1

𝜋 WB11 Transformations of sin graph Sketch the graph of 𝒔𝒊𝒏 𝒙+ 𝝅 𝟐 y = sinθ 1 𝜋 2 𝜋 3𝜋 2 2𝜋 y -1 Sketch the graph of a) 𝐬𝐢𝐧 𝒙+ 𝝅 𝟐 b) 𝒔𝒊𝒏 𝒙 −𝟐 c) − 𝒔𝒊𝒏 𝟐𝒙 𝒔𝒊𝒏 𝒙+ 𝝅 𝟐 − 𝒔𝒊𝒏 𝟐𝒙 𝒔𝒊𝒏 𝒙 −𝟐 −𝟑 −𝟐

WB12b double transformations (ax+b) 𝑦 =2 cos 𝜃+ 3𝜋 2 y 1 -1 y = cosθ -2π -3π 2 -π -π 2 2π 3π 2 π π 2 𝑦 = cos 𝜃+ 3𝜋 2 Sketch the graph of a) 2cos 𝑥+ 3𝜋 2 Transformations in order are i) shift − 3𝜋 2 in the x direction ii) Stretch ×2 in the y direction

𝜋 WB12b double transformations (ax+b) y = sinθ 𝜋 2 2𝜋 3𝜋 2 2𝜋 y -1 Sketch the graph of b) sin 2𝑥−𝜋 𝒔𝒊𝒏 𝒙− 𝝅 𝟐 𝒔𝒊𝒏 𝟐 𝒙− 𝝅 𝟐 =sin 2 𝑥− 𝜋 2 Transformations in order are i) shift + 𝜋 2 in the x direction ii) Stretch × 1 2 in the y direction

WB12c double transformations (ax+b) y=tan 1 2 𝑥+ 𝜋 2 Sketch the graph of c) tan 𝑥 2 + 𝜋 4 y=tan 𝑥+ 𝜋 2 y = tanθ 1 -1 -2π -3π 2 -π -π 2 2π 3π 2 π π 2 =tan 1 2 𝑥+ 𝜋 2 Transformations in order are i) shift − 𝜋 2 in the x direction ii) Stretch ×2 in the x direction

For any angle ϴ, sin(- ϴ) = - sinϴ cos(- ϴ) = cosϴ tan(- ϴ) = - tanϴ Reminder Symmetry of trig graphs 1 -1 𝜋 2 𝜋 3𝜋 2 2𝜋 y - 𝜋 2 - 𝜋 −2𝜋 - 3𝜋 2 y = sinθ y = cosθ y = tanθ For any angle ϴ, sin(- ϴ) = - sinϴ cos(- ϴ) = cosϴ tan(- ϴ) = - tanϴ Complete: sin − 𝝅 𝟐 = cos − 𝝅 𝟔 = tan − 𝝅 𝟑 =

Trig equations Solve trig equations in radians the same way as in degrees Rearrange to sin / cos / tan = Use arcsin / arccos / arctan on calculator to get a first answer in a list Use a graph to list all the answers in the given range Sometimes complete the solution with more inverses ( e.g. when its cos 3𝑥+ 𝜋 2 you do the operations − 𝜋 2 and ÷3 to your list

𝑥=− 𝜋 6 , 𝜋 6 , …. . What do you notice? WB13 Solve sin 𝑥+ 𝜋 2 = 3 2 in the range 0≤𝑥≤2𝜋 y = sinθ 1 𝜋 2 𝜋 3𝜋 2 2𝜋 y -1 𝑥+ 𝜋 2 = arcsin 3 2 = 𝜋 3 , 2𝜋 3 , …. 𝜋 3 2𝜋 3 7𝜋 3 𝑥= 𝜋 3 − 𝜋 2 , 2𝜋 3 − 𝜋 2 , …. 𝑥=− 𝜋 6 , 𝜋 6 , …. . What do you notice? The other answer in range is 7𝜋 3 − 𝜋 2 = 11𝜋 6 The final answers are 𝑥= 𝜋 6 , 11𝜋 6 check they work with your calculator

WB14 Solve each equation for x in the interval 0≤𝑥≤2𝜋 giving your answers in terms of π 𝑎) 𝑥 2 = 𝜋 4 , 3𝜋 4 , …. . 𝑥= 𝜋 2 , 3𝜋 2 , 𝑏) cos 𝑥+𝜋 = 2 2 𝑏) 𝑥+π= 𝜋 4 , 7𝜋 4 , 9𝜋 4 …. . 𝑥= 3𝜋 4 , 5𝜋 4 , 𝑐) tan 𝑥 2 − 𝜋 4 = 3 c) 𝑥 2 − π 4 = 𝜋 3 , 4𝜋 3 , …. . 𝑥= 7𝜋 6 , only

WB15 Solve each equation for x in the interval 0≤𝑥≤2𝜋 giving your answers to 2 decimal places 𝑎) sin 2𝑥=0.6 𝑎) 2𝑥= 0.644, 2.498, 6.927, 8.781 …. . 𝑥=0.32, 1.25, 3.46, 4.39 𝑏) cos 𝑥−0.3 =0.8 𝑥−0.3=−0.644, 0.644, 5.640, 6.927 …. . 𝑥=, 0.94, 5.94 𝑏) 2 cos 𝑥−0.3 =1.6 𝑐) 3−2 tan 𝑥− 𝜋 3 =0 c) tan 𝑥− 𝜋 3 = 3 2 𝑥− 𝜋 3 =0.9828, 4.1244 , …. . 𝑥= 2.03, 5.17

sin 𝑥 =+ 2 2 gives 𝑥= 𝜋 4 , 3𝜋 4 𝑠𝑖𝑛 2 𝑥= 1 2 sin 𝑥 =± 2 2 WB16 Solve each equation for θ in the interval −𝜋≤𝜃≤𝜋 giving your answers in terms of π 𝑎) 2 𝑠𝑖𝑛 2 𝑥 =1 sin 𝑥 =+ 2 2 gives 𝑥= 𝜋 4 , 3𝜋 4 𝑠𝑖𝑛 2 𝑥= 1 2 sin 𝑥 =± 2 2 sin 𝑥 =− 2 2 gives 𝑥=− 𝜋 4 ,− 3𝜋 4 𝑏) 4 𝑐𝑜𝑠 2 𝑥+4 cos 𝑥 +1=0 cos 𝑥 =−1 gives 𝑥=−𝜋, 𝜋 cos 𝑥 +1 2 cos 𝑥 −1 =0 cos 𝑥 = −1 , 1 2 sin 𝑥 = 1 2 gives 𝑥=− 𝜋 3 , 𝜋 3

cos 𝑥 = −(−1)± (−1) 2 −4(3)(−1) 2(3) WB17a Solve each equation for θ in the interval −𝜋≤𝜃≤𝜋 giving your answers to 3 significant figures 𝑎) 2 cos 𝑥 −1 = sin 𝑥 tan 𝑥 cos 𝑥 =−0.4343 gives 𝑥=−2.02, 2.02 2 cos 𝑥 −1=sin x sin x cos x 2 𝑐𝑜𝑠 2 𝑥 − cos 𝑥= 𝑠𝑖𝑛 2 𝑥 sin 𝑥 =− 2 2 gives 𝑥=−0.696, 0.696 2 𝑐𝑜𝑠 2 𝑥 − cos 𝑥=1− 𝑐𝑜𝑠 2 𝑥 3 𝑐𝑜𝑠 2 𝑥 − cos 𝑥 −1=0 cos 𝑥 = −(−1)± (−1) 2 −4(3)(−1) 2(3) cos 𝑥 = −0.4343, 0.7676

𝑏) 𝑐𝑜𝑠 2 𝑥−3 sin 𝑥 − 𝑠𝑖𝑛 2 𝑥=2 sin 𝑥 =− 1 2 gives 𝑥=− 5𝜋 6 ,− 𝜋 6 WB17b Solve each equation for θ in the interval −𝜋≤𝜃≤𝜋 giving your answers to 3 significant figures 𝑏) 𝑐𝑜𝑠 2 𝑥−3 sin 𝑥 − 𝑠𝑖𝑛 2 𝑥=2 sin 𝑥 =− 1 2 gives 𝑥=− 5𝜋 6 ,− 𝜋 6 (1− 𝑠𝑖𝑛 2 𝑥)−3 sin x− 𝑠𝑖𝑛 2 𝑥=2 0 = 2 𝑠𝑖𝑛 2 𝑥+3 sin 𝑥+1 sin 𝑥 =−1 gives 𝑥=− 𝜋 2 2 sin 𝑥 +1 sin 𝑥 +1 =0 sin 𝑥 − 1 2 , -1

WB18 Sketch the curve 𝑦 = 2cos⁡𝑥° for x in the interval 0≤𝑥≤2𝜋 WB18 Sketch the curve 𝑦 = 2cos⁡𝑥° for x in the interval 0≤𝑥≤2𝜋. Sketch on the same diagram the curve 𝑦 = cos⁡(𝑥 − 𝜋 3 ) . use your graph to find the values of x in the interval 0≤𝑥≤2𝜋 for which 2 cos 𝑥 =cos⁡ 𝑥 − 𝜋 3 From the graph 2 cos 𝑥 =cos⁡ 𝑥 − 𝜋 3 When y = 1 and y=-1 𝐴 𝑖𝑠 𝑝𝑜𝑖𝑛𝑡 𝜋 3 , 1 B 𝑖𝑠 𝑝𝑜𝑖𝑛𝑡 4𝜋 3 , −1

One thing to improve is – KUS objectives BAT know the key features of trig graphs using radians BAT solve trig equations using radians self-assess One thing learned is – One thing to improve is –

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