Trigonometric Functions 6C
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1 Trigonometric Functions 6C a b c d e sin 3 q æ ö ø 4 tan 6 q 4 æ ö tanq ø cos q æ ö ø 3 cosec 3 q - sin q sin q cos q sin q (using sin q + cos q ) So - sin q sin q æ ö ø 6 4cot 6 q sec q cot q secq cos 4 q cos 4 q cos 5 q æ ö ø 5 sec 5 q b tan x - tan x - cot x - c 3 3sin x cos x (multiply by ) 3 cos x sin x (divide by sin x) æ ö ø cot x 3 3 cot x ± 3 3 a cotq f g h cosec 3 q cotq secq sin 3 q sin 4 q sin q æ ö ø tanq (tanq) cosec q cot q cosec q tan q sin q sin q cos q a æ ö ø 3 sec 3 q (divide by ) b tanq cotq tanq tanq c tanq cosecq secq d (cotq + tanq) æ + ö ø cos q + sin q e sin 3 xcosecx + cos 3 xsec x sin 3 x + cos3 x sin x + cos x 5 cot x (divide by 4) 4 Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.
2 3 f sec A- sec Asin A sec A(- sin A) (factorise) cos A cos A (using sin A + cos A ) cos A g sec xcos 5 x + cot xcosec xsin 4 x cos x cos5 x + sin4 x cos 3 x + sin x (cos x + sin x) (since cos x + sin x ) 4 a LHS + tanq + cos q + sin q (using sin q + cos q ) secq RHS b LHS cotq + tanq + cos q + sin q cosecq secq RHS c LHS cosecq sin q cos q cotq RHS d LHS (- )(+ sec x) - + sec x - sec x (multiplying out) sec x - (as sec x ) - - cos x tan x RHS e LHS cos x + (- ) (- ) cos x + (- + sin x) (- ) - (- ) (using sin x + cos x ) (- ) (- ) (factorising) sec x RHS f LHS + cotq + tanq tanq+ tanq tanq + tanq + tanq + tanq RHS Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.
3 5 a secq Calculator value is q 45 is positive q is in st and 4th quadrants Solutions are 45, 35 b cosecq Calculator value is q ( d.p.) is negative q is in 3rd and 4th quadrants Solutions are 99, 34 (3 s.f.) c 5cotq - cotq - 5 d cosecq Calculator value is q 30 is positive q is in st and nd quadrants Solutions are 30, 50 e 3sec q 4 sec q 4 3 cos q 3 4 ± 3 Calculator value for 3 is q 30 As is ±, q is in all four quadrants Solutions are 30, 50, 0, 330 f 5 3cotq 5 3 Do not cancel on each side. Multiply through by (5-3) 0 (factorise) tanq - 5 Calculator value is q ( d.p.) tanq is negative q is in nd and 4th quadrants So 0 or 3 5 When 0, q 90, 70 When 3 5, q 36.9, 43 (3 s.f.) Solutions are 36.9, 90, 43, 70 Solutions are, 9 (3 s.f.) Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 3
4 5 g cot q -8tanq 0 tan q - 8tanq 0-8tan 3 q 0 8tan 3 q tan 3 q 8 b secq -3-3 Calculator value is q 09 (3 s.f.) is negative q is in nd and 3rd quadrants tanq Calculator value is q 6.57 ( d.p.) tanq is positive q is in st and 3rd quadrants Solutions are 6.57 and ( ) So solutions are 6.6, 07 (3 s.f.) Solutions are 09, -09 (3 s.f.) h cosecq sin q ± Calculator value for is q 45 c cotq 3.45 tanq 3.45 tanq (4 d.p.) 3.45 Calculator value is q 6.6 ( d.p.) tanq is positive q is in st and 3rd quadrants Solutions are in all four quadrants Solutions are 45, 35, 5, 35 6 a cosecq q 90 Solutions are 6. and ( ) So solutions are 6., -64 (3 s.f.) Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 4
5 6 d cosec q - 3cosecq0 cosecq(cosecq - 3) 0 (factorise) cosecq 0 or cosecq 3 3 cosecq 0 has no solutions Calculator value for is q q is in st and nd quadrants Solutions are 4.8, (80-4.8) So solutions are 4.8, 38 (3 s.f.) g cosecq 4 4 Remember that solutions are required in the interval -80 q 80 So q 360 Calculator value for 4 is q 4.48 ( d.p.) is positive q is in st and nd quadrants e secq cos q ± Calculator value for is q 45 q is in all quadrants, but remember that solutions required for -80 q 80 Solutions are ± 45, ±35 q , , 4.48, 65.5 q -97., - 7.8, 7.4, , - 97., 7.4, 8.8 (3 s.f.) f 3cotq 3 3 sin q 3 (- cos q) (use sin q + cos q ) cos q ( -)( + ) 0 or - As - has no solutions, Solutions are ± 60 Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 5
6 6 h cot q - cotq -5 0 As this quadratic in cotq does not factorise, use the quadratic formula cotq -b ± b - 4ac a (You could change cotq to tanq and work with the quadratic 5tan q + tanq - 0) ± 4 So cotq ,.8508 (4 d.p.) So tanq , (4 d.p.) The calculator value for tanq is q ( d.p.) b cotq - 3 tanq - 3 Calculator solution is - p 6 ( you should know that tan p 6 3 ) - p is not in the interval 6 Solutions are p - p 6, p - p 6 5p 6, p 6 Solutions are , 43 (3 s.f.). The calculator value for tanq is q 8.38 ( d.p.) c cosec q 3 3 sin q Remember that 0 q p so 0 q p First solution for sin q 3 is q p 3 Solutions are 8.4, ( ) Total set of solutions is -5, , 8.4, 43 (3 s.f.) 7 a secq - - q p (refer to graph of y ) So q p 3, p 3 q p 3, 4p 3 Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 6
7 7 d secq tanq ( ¹ 0) Solutions are p 4, 3p 4 8 a In the right-angled triangle ABD AB AD AD 6 6secq In the right-angled triangle ACB AC AB AC 6 CD AD - AC 6secq - 6 6(secq - ) 9 a b As 6 6secq cos q 3cos q (3 -)( + 3) 0 3 as ¹ -3 From (a) AC cm cosec x - cot x cosec x b By part a equation becomes cosec x is positive, so x is in st and nd quadrants x p 6,5p 6 0 a tan x - - sin x (- ) cos x (- ) - (- ) sec x b Need to solve sec x - - which has no solutions. Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 7
8 + cot x + tan x cot x tan x 5 Calculator solution is.3 ( d.p.) tan x is positive, so x is in st and 3rd quadrants Solutions are.3, 9.3 ( d.p.) Pearson Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free. 8
h (1- sin 2 q)(1+ tan 2 q) j sec 4 q - 2sec 2 q tan 2 q + tan 4 q 2 cosec x =
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