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Trigonometry: Addition formula for Rcos (x+a) and R(x+a)
KUS objectives BAT write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only BAT solve equations of the form acosθ + bsinθ, or find their max/min value Starter:
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R sin (𝜃+ ∝) =𝑹 sin 𝜃 𝐜𝐨𝐬 ∝ +𝑹 cos 𝜃 𝐬𝐢𝐧 ∝
WB1 Show that 3 sin 𝜃 +4 cos 𝜃 can be expressed in the form 𝑅 sin 𝜃+∝ where 𝑅>0, 0< 𝛼< 𝜋 2 R sin (𝜃+ ∝) =𝑹 sin 𝜃 𝐜𝐨𝐬 ∝ +𝑹 cos 𝜃 𝐬𝐢𝐧 ∝ Compare each term – they must be equal! 𝟑 𝑠𝑖𝑛𝜃 − 𝟒 𝑐𝑜𝑠𝜃 α 𝑅 3 4 So in the triangle, the Hypotenuse is R… 𝑹 𝒔𝒊𝒏 ∝ =𝟔 𝑹 𝐜𝐨𝐬 ∝ =𝟑 → 𝑡𝑎𝑛 ∝ = 6 3 =2 𝑅= = 5 → ∝ =1.107 SO 𝟑 𝑠𝑖𝑛𝜃 − 𝟒 𝑐𝑜𝑠𝜃 =5 sin 𝜃+1.107
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R cos (𝜃− ∝) =𝑹 cos 𝜃 𝐜𝐨𝐬 ∝ +𝑹 sin 𝜃 𝐬𝐢𝐧 ∝
WB2: a) Express 2 cos 𝜃 +5 sin 𝜃 can be expressed in the form 𝑅 cos 𝜃−∝ where 𝑅>0, 0< 𝛼< 𝜋 2 b) Hence, solve 2 cos 𝜃 +5 sin 𝜃 =3 in the range 0<𝜃<2π R cos (𝜃− ∝) =𝑹 cos 𝜃 𝐜𝐨𝐬 ∝ +𝑹 sin 𝜃 𝐬𝐢𝐧 ∝ Compare each term carefully- get sin and cos in correct order 𝟐 cos 𝜃 𝟓 sin 𝜃 α 𝑅 2 5 So in the triangle, the Hypotenuse is R… 𝑹 𝒔𝒊𝒏 ∝ =𝟓 𝑹 𝐜𝐨𝐬 ∝ =𝟐 → 𝑡𝑎𝑛 ∝ = 5 2 𝑅= = 29 → ∝ =1.190 SO 𝟐 cos 𝜃 +𝟓 sin 𝜃 = sin 𝜃−1.190 cos 𝜃−1.190 = 𝜃− = 0.980, 5.303 𝜃 = 2.17, 6.49
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Compare each term – they must be equal!
WB 3a a) Show that you can express sin 𝜃 − 3 cos 𝜃 in the form 𝑅 sin 𝜃+∝ where 𝑅>0, 0< 𝛼<90 b) Hence, sketch the graph of y= sin 𝑥 − 3 cos 𝑥 R sin (𝜃+ ∝) =𝑹 sin 𝜃 𝐜𝐨𝐬 ∝ +𝑹 cos 𝜃 𝐬𝐢𝐧 ∝ Compare each term – they must be equal! 𝟏 sin 𝜃 − 𝟑 𝑐𝑜𝑠𝜃 → 𝑡𝑎𝑛 ∝ = 𝟑 1 = 𝟑 𝑅= = 2 𝑹 𝒔𝒊𝒏 ∝ = 𝟑 𝑹 𝐜𝐨𝐬 ∝ =𝟏 → ∝ = 𝜋 3 SO sin 𝜃 − 𝟑 𝑐𝑜𝑠𝜃 =2 sin 𝜃+ 𝜋 3
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WB3b: b) Sketch the graph of: 𝑠𝑖𝑛𝑥− 3 𝑐𝑜𝑠𝑥 2sin 𝑥− 𝜋 3
𝑠𝑖𝑛𝑥− 3 𝑐𝑜𝑠𝑥 2sin 𝑥− 𝜋 3 = Sketch the graph of: 1 y=sin𝑥 Start out with sinx -1 π/2 π 3π/2 2π y=sin 𝑥− 𝜋 3 1 Translate π/3 units right -1 π/3 π/2 π 4π/3 3π/2 2π 2 y=2sin 𝑥− 𝜋 3 Vertical stretch, scale factor 2 1 -1 π/3 π/2 π 4π/3 3π/2 2π -2 At the y-intercept, x = 0 2sin − 𝜋 3 =− 3
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Remember to get the + or – signs the correct way round!
Which one should be used? 𝑎𝑠𝑖𝑛𝜃±𝑏𝑐𝑜𝑠𝜃 𝑅𝑠𝑖𝑛 𝜃±𝛼 𝑎𝑐𝑜𝑠𝜃±𝑏𝑠𝑖𝑛𝜃 𝑅𝑐𝑜𝑠 𝜃∓𝛼 Whichever ratio is at the start, change the expression into a function of that (This makes solving problems easier) Remember to get the + or – signs the correct way round!
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Which addition formula should we use?
WB4: Solve the equation 9𝑠𝑖𝑛𝜃 −6𝑐𝑜𝑠𝜃=7 𝑓𝑜𝑟 0<𝜃<2π Which addition formula should we use? R sin (𝜃− ∝) =𝑹 sin 𝜃 𝐜𝐨𝐬 ∝ −𝑹 cos 𝜃 𝐬𝐢𝐧 ∝ 𝟗 𝑠𝑖𝑛𝜃 − 𝟔 𝑐𝑜𝑠𝜃 =7 𝑹 𝒔𝒊𝒏 ∝ =𝟔 𝑹 𝐜𝐨𝐬 ∝ =𝟗 → 𝑡𝑎𝑛 ∝ = 6 9 𝑅= =3 13 → ∝ =0.588 So we have 𝑠𝑖𝑛 𝜃−5.88 =7 → 𝑠𝑖𝑛 𝜃−0.588 = → 𝜃− =0.704, 2.438 → 𝜃 =1.292, 3.026
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Which addition formula should we use?
WB5: Find the maximum value of the value of the expression 2sinx + 3cosx, and find the value of x (0<x<2π) at which the maximum occurs Which addition formula should we use? R sin (𝑥+ ∝) =𝑹 sin 𝑥 𝐜𝐨𝐬 ∝ + 𝑹 cos 𝑥 𝐬𝐢𝐧 ∝ 𝟐 sin 𝑥 𝟑 cos 𝑥 → 𝑡𝑎𝑛 ∝ = 3 2 𝑹 𝒔𝒊𝒏 ∝ =𝟑 𝑹 𝐜𝐨𝐬 ∝ =𝟐 𝑅= = 13 → ∝ =0.983 SO 𝟐 sin 𝑥 + 𝟑 cos 𝑥 = sin 𝜃+0.983 This has a maximum when 𝜃+0.983= 𝜋 2 → 𝜃= in the range 0<𝑥<2𝜋 → cos 𝜃 = 13
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R cos (𝜃− ∝) =𝑹 cos 𝜃 𝐜𝐨𝐬 ∝ −𝑹 sin 𝜃 𝐬𝐢𝐧 ∝
WB6: a) Express 12 cos 𝜃 +5 sin 𝜃 in the form 𝑅 cos 𝜃−∝ where 𝑅>0, 0< 𝛼<90 b) Hence, find the maximum value of 12 cos 𝜃 +5 sin 𝜃 and the smallest positive value of at which it arises R cos (𝜃− ∝) =𝑹 cos 𝜃 𝐜𝐨𝐬 ∝ −𝑹 sin 𝜃 𝐬𝐢𝐧 ∝ 𝟏𝟐 cos 𝜃 𝟓 sin 𝜃 → 𝑡𝑎𝑛 ∝ = 5 12 𝑹 𝒔𝒊𝒏 ∝ =𝟓 𝑹 𝐜𝐨𝐬 ∝ =𝟏𝟐 𝑅= =13 → ∝ =0.395 SO 𝟏𝟐 cos 𝜃 + 𝟓 sin 𝜃 =13 cos 𝜃−0.935 This has a maximum when 𝜃=0.935 → 13 cos 𝜃− =13
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Which addition formula should we use?
WB7: Express 4𝑐𝑜𝑠𝑥+3𝑠𝑖𝑛𝑥 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝑟𝑐𝑜𝑠(𝑥−,) 𝑤ℎ𝑒𝑟𝑒 𝑅>0 𝑎𝑛𝑑 0<∝< 𝜋 2 . Hence solve 4 𝑐𝑜𝑠𝑥+3𝑠𝑖𝑛𝑥=2 𝑓𝑜𝑟 0<𝑥<𝜋, giving your answers to 2.d.p Which addition formula should we use? R cos (𝑥−∝) =𝑹 cos 𝑥 𝐜𝐨𝐬 ∝ + 𝑹 sin 𝑥 𝐬𝐢𝐧 ∝ 𝟒 cos 𝑥 𝟑 sin 𝑥 → 𝑡𝑎𝑛 ∝ = 3 4 𝑹 𝒔𝒊𝒏 ∝ =𝟑 𝑹 𝐜𝐨𝐬 ∝ =𝟒 𝑅= =5 → ∝ =0.643 SO 𝟒 cos 𝑥 + 𝟑 sin 𝑥 =5 cos 𝑥− =2 →cos 𝑥− = 2 5 → 𝑥− =1.159, 5.124, …. →𝑥 =1.80, only in the given range
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self-assess using: R / A / G ‘I am now able to ____ .
KUS objectives BAT write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only BAT solve equations of the form acosθ + bsinθ, or find their max/min value self-assess using: R / A / G ‘I am now able to ____ . To improve I need to be able to ____’
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