Trig Equations

f(x)= cos 2𝑥, 0°<𝑥≤360° g(x)=tan 𝑥−60 , −360°<𝑥≤0° Trigonometry KUS objectives BAT rearrange and solve trig equations Starter: sketch these graphs f(x)= cos 2𝑥, 0°<𝑥≤360° g(x)=tan 𝑥−60 , −360°<𝑥≤0° h x =−sin 𝑥+45 , −180°<𝑥≤180° Check using Desmos / geogebra

Solve sin 𝜃 =0.5 in the interval 0≤ 𝜃 ≤360° WB21 Solve sin 𝜃 =0.5 in the interval 0≤ 𝜃 ≤360° One thing you should pay careful attention to is the range the answers can be within, Use Sin-1 This will give you one answer 0.5 y = Sinθ 90 180 270 360 30 150 The infinite set of solutions for −∞≤ 𝜃 ≤∞° , would be 𝜃=30±360𝑛 𝜃=150±360𝑛

Solve 5sin 𝜃 =−2 in the interval 0≤ 𝜃 ≤360° WB22 Solve 5sin 𝜃 =−2 in the interval 0≤ 𝜃 ≤360° Divide by 5 Use Sin-1 Not within the range. You can add 360° to obtain an equivalent value 203.6 336.4 90 180 270 360 y = Sinθ -0.4 The infinite set of solutions for −∞≤ 𝜃 ≤∞° , would be … ? 𝜃=203.6±360𝑛 𝜃=336.4±360𝑛

tan 𝜃 = 9 2 𝜃=77.5 Solve 2 tan 𝜃 =−9 in the interval −90°≤ 𝜃 ≤90° WB23 Solve 2 tan 𝜃 =−9 in the interval −90°≤ 𝜃 ≤90° 90º 180º 270º 360º -90º -180º -360º 1 -1 -270º θ tan 𝜃 = 9 2 Divide by 2 Use Tan-1 𝜃=77.5 There is only one solution if −90°≤ 𝜃 ≤90° The infinite set of solutions for −∞≤ 𝜃 ≤∞° , would be … ? 𝜃=77.5±180𝑛

The infinite set of solutions WB24 Solve cos 𝜃 =0.5 in the interval −90≤ 𝜃 ≤ 270 90 y = Cosθ Y=0.5 270 360 180 -90 cos 𝜃 = 1 2 Use Tan-1 𝜃=…60, 300, … The infinite set of solutions for −∞≤ 𝜃 ≤∞° , would be … ? Now consider −90≤ 𝜃 ≤ 270 𝜃=±60±360𝑛 𝜃=−60, 𝑜𝑟 𝜃=60

2sin 𝜃=1.6, 0≤𝜃≤180 𝑡𝑎𝑛𝜃=2, 0≤𝜃≤360 cos 𝜃 = 2 5 , 0≤𝜃≤360 Practice 1 Solve these equations, give exact answers or round to 3 s.f. 2sin 𝜃=1.6, 0≤𝜃≤180 𝑡𝑎𝑛𝜃=2, 0≤𝜃≤360 cos 𝜃 = 2 5 , 0≤𝜃≤360 𝑡𝑎𝑛𝜃+8=11, −180≤𝜃≤180 cos 𝜃 = 1 5 , −360≤𝜃≤360 3 sin 𝜃 =4, 0≤𝜃≤360 3 tan 𝜃 =1, 0≤𝜃≤360 −3 sin 𝜃 =8, 0≤𝜃≤360 cos 𝜃 =− 2 5 , 0≤𝜃≤360 3 𝑡𝑎𝑛𝜃=−6, 0≤𝜃≤720 sin 𝜃 =− 1 2 , 0≤𝜃≤360 cos 𝜃 =− 1 2 , −360≤𝜃≤0

Solve cos 2𝜃 =−1 in the interval 0≤ 𝜃 ≤360° WB25 to solve equations in the form Sin/Cos/Tan(aθ + b) = k Solve cos 2𝜃 =−1 in the interval 0≤ 𝜃 ≤360° 1) Work out the acceptable interval for 2θ Multiply by 2 Solve using Cos-1 2) Work out one possible answer as before. Find all values in the standard 0 – 360 range y = Cosθ 3) Add/Subtract 360 to these values until you have all the answers within the 2θ range 90 180 270 360 -1 180 4) These answers are for 2θ. Undo them to find values for θ itself

1) Work out the acceptable interval for (θ-60) WB26 to solve equations in the form Sin/Cos/Tan(aθ + b) = k Solve tan (𝜃−60) =−1 in the interval 0≤ 𝜃 ≤360° 1) Work out the acceptable interval for (θ-60) Subtract 60 −60≤ 𝜃≤300 2) Work out one possible answer as before. Find all values in the standard 0 – 360 range tan (𝜃−60) =−1 Solve using tan-1 𝜃−60=−45 3) Add/Subtract 180 to these values until you have all the answers within the (20 - θ) range 90º 180º 270º 360º -90º -180º -360º 1 -1 -270º θ 4) These answers are for (20 – θ). Undo this to find values for θ itself 𝜃−60=−45, 135, 315, 𝜃= 15, 195, Two answers between 0≤ 𝜃 ≤360°

1) Work out the acceptable interval for (2θ – 35) WB27 to solve equations in the form Sin/Cos/Tan(aθ + b) = k Solve sin (2𝜃−35) =−1 in the interval−180°≤ 𝜃 ≤180° Multiply by 2. Subtract 35 1) Work out the acceptable interval for (2θ – 35) Solve using Sin-1 2) Work out one possible answer as before. Find all values in the standard 0 – 360 range y = Sinθ 3) Add/Subtract 360 to these values until you have all the answers within the (2θ - 35) range 90 180 270 360 -1 270 Adding/Subtracting 360 to the value we worked out (staying within the range) 4) These answers are for (2θ – 35). Undo this to find values for θ itself Add 35, Divide by 2

1) Work out the acceptable interval for (20 – θ) WB28 to solve equations in the form Sin/Cos/Tan(aθ + b) = k Solve tan (20−𝜃) =3 in the interval−180°≤ 𝜃 ≤180° 1) Work out the acceptable interval for (20 – θ) Multiply by -1 Solve using tan-1 Add 20 2) Work out one possible answer as before. Find all values in the standard 0 – 360 range ‘Turn round’ 71.6 251.6 3) Add/Subtract 180 to these values until you have all the answers within the (20 - θ) range 3 y = Tanθ 90 180 270 360 Adding/Subtracting 180 to the values we worked out (staying within the range) 4) These answers are for (20 – θ). Undo this to find values for θ itself Subtract 20 Multiply by -1

Practice 2 Solve these equations sin 2𝜃= 1 3 , 0≤𝜃≤180 sin 2𝜃−30 = 1 3 , 0≤𝜃≤180 cos (𝜃+180) = 1 4 , 0≤𝜃≤360 cos (3𝜃−10) = 1 4 , 0≤𝜃≤360 tan 𝑥 2 =0. 839 0≤𝜃≤360 tan 1 2 𝜃+20 =2+ 3 , 0≤𝜃≤360 sin (𝜃−40)=0.94 0≤𝜃≤360 sin 5𝜃−100 =0.94 −90≤𝜃≤180 cos (4𝜃) =0.174 , 0≤𝜃≤90 cos 1 3 𝑥−10 =0.174 , −360≤𝜃≤0 tan 𝜃−45 =2− 3 0≤𝜃≤360 tan 2𝑥−15 =0.364 , −90≤𝜃≤90

One thing to improve is – KUS objectives BAT rearrange and solve trig equations self-assess One thing learned is – One thing to improve is –

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