Trig Review

4.

(a)

Solve, for 0   x  360 , the equation

2 cos ec x  1  3 sin x (b)

[4]

Find the values of y between 0 and 6 for which

3 tan 2 y  1  1  0 .

[4]

7

(a)

Find all the angles between 0 and 360 which satisfy the equations (i)

5 cos 2 x  1

[3]

(ii)

4 cosec 2 y  7  cot 2 y  2 cot y

[4]

1 Sketch the curve of y  3 cos x for 0  x  2 . 2 1 x Hence, find the number of solutions to the equation cos x   1 . 2 

(b)

[5]

2

1  sin 2 x  cos 2 x  2 cos x . cos x  sin x

(a)

Prove that

(b)

Find all the angles, in terms of , for 0  x  5 which satisfy the equation sin 2 x  sin( x  4) .

[4]

[6]

3

If A is an obtuse angle whose sine is p and B is an acute angle whose cosine is q, find, in terms of p and q, the values of (i)

tan 2 A ,

(ii)

cos( A  B) ,

(iii)

sin

B . 2

[7]

7

Solve the equation cos 3  sin   cos   0 for 0    360 .

[6]

9

9 20 and cos B   and A and B are in the same quadrant, find, 41 29 without calculating the values of A and of B , the exact value of (i) tan  A  B  and   (ii) sec   A  [6] 2 

If sin A 

2.

x The curve, y  a cos   c is defined for 0  x  4 , where a, b and c are b positive integers. Given that the amplitude of y is 5 and that the period of y is 4 . (a) State the value of a and of b. [2] Given that the minimum value of y is  7 , (b) state the value of c; and [1] (c) sketch the graph of y, indicating the coordinates of any maximum or minimum points. [3] x   State the range of values of k for which the equation a cos   c  k has 2 b unique solutions. [1]

11.

A

 F

4 cm B

The diagram shows two right-angled triangles OBC and OAB. The sides OB and AB are of length 10 cm and 4 cm respectively. OB is inclined at an angle  to OC, and BAD   . The line AD is perpendicular to OC, and intersects OB at the point E. BF is parallel to CD. All angles are measured in degrees.

E

10 cm

 O

D

C

(a)

Show that OD  10 cos  4 sin  .

[3]

(b)

Express OD in the form R cos(   ) .

[2]

(c)

Express AD in terms of R,  and  .

[2]

(d)

Show that the area of triangle OAD is 29 sin 2(   ) . Hence find the maximum value of the area of the triangle OAD and the corresponding value of  . [7]

2.

1  cos 2 x  sin 2 x  m cos x . cos x  sin x

(a)

Find the value of m for which

(b)

2 5 If sin A  cos A  , show that sin A cos A  . 3 18

[3]

Hence find the value of sec A  cos ec A . [4]

6.

Given that g ()  12cos   5sin  , (i) (ii)

express g () in the form R cos(  ) where R 0 and 0    90 . [3] Find the greatest and least values of g () and the corresponding values of  in the interval between 00 and 3600 for which the greatest and least values occur. [4]

(b) Prove the identity sin 2  tan  cos 2  tan  .

[3] 

 1 Hence, without using a calculator, show that tan  67   2  1 .  2

[3]

1

(a)

(b)

Find all the angles between 0o and 360o which satisfy the equation 2 cos 2 x  3  4 sin x Find all the angles between 0 and 4 which satisfy the equation   sin  x    3 cos x 3 

[4]

[4]

Given that 5 sin x  12 cos x  R sin( x   ) , where R > 0 and  is acute, find

10

(i)

the values of R and  .

[2]

(ii)

Hence, find the maximum value of 5 sin x  12 cos x  3 and the corresponding value of x for 0   x  360  .

[3]

(iii)

Find the smallest positive angle x such that this angle x satisfies 5 sin x  4  12 cos x .

[3]

11 (a) (b)

Prove the identity cos x sin x  cos 3 x cos ec x  cot x .

[3]

(i)

Prove that sin 3  sin   4 sin  cos 2  .

[3]

(ii)

Hence, find all angles  , 0     , for which sin 3  sin   0 .

[4]

12 (a)

(i)

The graph of y  3 cos 4 x  1 is defined for 0  x   . State the period of y.

[1]

(ii)

State the amplitude of y.

[1]

(iii)

Sketch the graph of y.

[3]

(iv) (v)

(b)

On the diagram drawn in part (iii), sketch the graph of 2x y  2 for 0  x   .  State the number of solutions, for 0  x   , of the equation 3 cos 4 x  3  2 x .

[1]

[2]

[3] Sketch the graph of y  ln x 2 . Insert on your sketch the additional graph required to illustrate [2] the graphical solution of the equation ln x  2  x .