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1.7.21 Use identities to solve equations (A-level only)

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Question 11
a.

Show that the equation

3cos⁡ϕ(tan⁡ϕsin⁡ϕ+1)=8cos⁡ϕ+1 3 \cos \phi ( \tan \phi \sin \phi + 1 ) = 8 \cos \phi + 1 3cosϕ(tanϕsinϕ+1)=8cosϕ+1

can be written in the form

3cos⁡2ϕ+5cos⁡ϕ−2=0 3 \cos^2 \phi + 5 \cos \phi - 2 = 0 3cos2ϕ+5cosϕ−2=0
[3]
b.

A mechanical sensor detects oscillations described by the phase equation

3cos⁡3x(tan⁡3xsin⁡3x+1)=8cos⁡3x+1 3 \cos 3x ( \tan 3x \sin 3x + 1 ) = 8 \cos 3x + 1 3cos3x(tan3xsin3x+1)=8cos3x+1

where x x\,x represents the angular position in degrees. Hence solve this equation for 0∘<x<180∘0^\circ < x < 180^\circ0∘<x<180∘, giving your answers to one decimal place.

[4]

1.7.21 Use identities to solve equations (A-level only) Questions

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