Archbishop Tenison's School

C3 Trigonometry

340 minutes

283 signifies

1 . (a)Prove that for any values of x,

sin x + bad thing (60° - x) º sin (60° + x).

(4)

(b)Given that trouble 84° -- sin 36° = trouble a°, consider the exact value of the severe angle a. (2)

(c)Solve the equation

some sin 2x + bad thing (60° - 2x) sama dengan sin (60° + 2x) – you

for values of by in the interval 0 £ x < 360°, giving your answers to one fraccion place. (5)

(Total 10 marks)

installment payments on your On individual diagrams, design the figure with equations

(a)y = arcsin x, -1 £ x £ 1,

(b)y = sec x, - £ back button £, declaring the runs of the end points of the curves in each case.

(4)

Utilize trapezium regulation with five equally spread ordinates to estimate the location of the region bounded by the curve with equation con = sec x, the x-axis and the lines by = and x = -, giving your reply to two decimal places. (4)

(Total almost eight marks)

three or more. Find, supplying your answers to two quebrado places, the values of w, by, y and z that (a)e-w = 4,

(2)

(b)arctan x = you,

(2)

(c)ln (y & 1) – ln con = zero. 85

(4)

(d)cos z . + desprovisto z sama dengan, -p < z < p.

(5)

(Total 13 marks)

four. (a)Using the formulae

desprovisto (A ± B) sama dengan sin A cos B ± cos A sin B,

cos (A ± B) sama dengan cos A cos W sin A sin B,

show that

(i)sin (A + B) – bad thing (A – B) = 2 cos A bad thing B,

(2)

(ii)cos (A – B) – cos (A & B) sama dengan 2 desprovisto A sin B.

(2)

(b)Use the above results to present that

= crib A.

(3)

Using the consequence of part (b) and the exact values of sin 60° and cos 60°,

(c)find an exact value pertaining to cot 75° in its easiest form.

(4)

(Total 10 marks)

a few. In a particular circuit the current, I amperes, is given simply by I = 4 trouble q – 3 cos q, queen > 0,

where q is a great angle associated with the volts.

Given that I actually = R sin (q - a), where Ur > zero and zero £ a < 360°,

(a)find the value of 3rd there�s r, and the worth of a to at least one decimal place. (4)

(b)Hence solve the equation four sin q – a few cos queen = three or more to find the principles of queen between zero and 360°.

(5)

(c)Write down the very best value pertaining to I.

(1)

(d)Find the value of queen between zero and 360° at which the greatest value of I happens. (2)

(Total 12 marks)

6. (a)Express sin times + Ö3 cos times in the kind R sin (x + a), wherever R > 0 and 0 < a < 90°. (4)

(b)Show which the equation sec x & Ö3 cosec x = 4 may be written inside the form trouble x & Ö3 cos x sama dengan 2 sin 2x.

(3)

(c)Deduce coming from parts (a) and (b) that sec x & Ö3 cosec x = 4 could be written in the form sin 2x – sin (x + 60°) = 0.

(1)

(d)Hence, using the personality sin Back button – desprovisto Y sama dengan 2 cos, or otherwise, discover the ideals of times in the time period 0 £ x £ 180°, for which sec x + Ö3 cosec back button = some. (5)

(Total 13 marks)

7. (i)Given that cos(x + 30)° = 3 cos(x – 30)°, provide evidence that tan x° = --. (5)

(ii)(a)Prove that º tan queen.

(3)

(b)Verify that q = 180° is a option of the equation sin 2q = a couple of – two cos 2q. (1)

(c)Using the result simply (a), or else, find the other two solutions, zero < q < 360°, of the equation sin 2q = 2 – 2 cos 2q. (4)

(Total 13 marks)

8. (a)Prove that

(4)

(b)Hence, or otherwise, demonstrate

tan2 = a few – 2Ö2.

(5)

(Total 9 marks)

9. (i)(a)Express (12 cos q – 5 sin q) in the form L cos (q + a), where R > zero and zero < a < 90°.

(4)

(b)Hence solve the equation

doze cos queen – your five sin queen = 5,

for zero < queen < 90°, giving the answer to one particular decimal place.

(3)

(ii)Solve

8 crib q – 3 color q sama dengan 2,

to get 0 < q < 90°, providing your reply to 1 quebrado place.

(5)

(Total doze marks)

15. (i)Given that sin back button =, how to use appropriate double angle formulation to find the actual value of sec two times. (4)

(ii)Prove that

cot 2x + cosec 2x º cot back button, (x ¹, n Î ).

(4)

(Total almost eight marks)

11.

This diagram shows a great isosceles triangle ABC with AB = AC = 4 cm and Ð BAC sama dengan 2q.

The mid-points of AB and AC are D and E respectively. Rectangle DEFG is attracted, with Farrenheit and G on BC. The edge of rectangle DEFG is usually P cm. (a)Show that DE = 4 bad thing q.

(2)

(b)Show that P sama dengan 8 sinq + 4 cosq.

(2)

(c)Express L in the...