00 (3) 2016 - 2017 H2 Maths Trigonometry Practice Questions (student)

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 1 of 5 National Junior College 2016 – 2017 H2 Mathematics Revision: Trigonometry Practice Questions Basic Mastery Questions Basic Trigonometric Functions 1. By using a right-angled triangle, show that (a) sin cos ,2    (b) cos sin ,2    (c) tan cot .2    Remark: Even though we have only justified the above results for an acute angle  , in fact these relationships hold for all values of . 2. It is given that 120 .A Without using a calculator, w rite down the smallest positive integer n such that each of the following trigonometric expression yields a positive value. Justify your answer clearly. (a) sin( ),nA (b) tan( ),nA (c) cos( ).nA Relationship between Basic Trigonometric Functions and their inverses 3. Find the exact value of   1cos tan 1 . 4. It is given that 1tan , x  where x is positive. (a) State the range of possible values of . (b) Find the following in terms of x: (i) cot , (ii) sec , (iii) cosec . Use of Formulae 5. It is given that 11tan and tan .23AB Evaluate    tan and tan 2A B A without the use of a calculator. 6. Express sin 3 cosxx in the form sin ( )Rx  , where 0R and 0 2  are constants whose exact values are to be determined. Trigonometric Equations 7. Write down the basic angle for 1cos . 2x Hence solve 1cos , 2x where 0 2 .x  8. Write down the basic angle for 1tan . 3 x Hence solve 1tan , 3 x where 0 2 .x  9. Solve sin 2 0, where 0 2 .4xx      

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 2 of 5 Sine and Cosine Rule 10. The diagrams above (not drawn to scale) show four triangles. Find the possible value(s) of (a) ,ABC (b) ,DF (c) HI, (d) .JLK Intermediate Level Questions Compound Angle Formulae 1. Given that 3sin 5A and 12cos , 13B prove that one possible value of cos( )AB is 33 ,65 and find all the other possible values. [J80/P1/4] Proving Trigonometric Identities 2. Prove the following identities: (a) sin 1 cosec ,1 cos 2 2 A AA (b) 2 2cos 2cos , 2 xx (c) 44cos sin cos 2 ,   (d) cos sin tan ,cos sin 4 xx xxx     (e) sin 5 sin cot 2 ,cos5 cos     (f) 1 sin 6sin 3 cos 3 .4 4 2 xxx              Trigonometric Equations 3. Solve sin 2 sin , where 0 2 .x x x    (Attempt this question using two approaches: Factor and Double angle formula). 4. Solve 2sin 2 sin 2 2, where 0 2 .x x x     Sine and Cosine Rule 5. The diagram (not drawn to scale) show triangle XYZ. Find the lengths of XY and YZ. 6. A triangle has side lengths 6 cm, 8 cm and 11 cm. Find the smallest angle in the triangle.

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 3 of 5 Overall 7. NJC/Mid Year 2006/Q6 (a) Prove the identity 322cos cos3 cos5 16cos sin       . [3] (b) Express 2 1f( ) cos sin 2 12     in the form of   1cos 2 , 2R  where and 0 , 2R    giving the exact values of R and . [3] (c) Hence give the range of values of   2 f( ) . [2] 8. Consider a triangle ABC with .abc (i) What can you conclude about the relative sizes of sin A, sin B and sin C? (ii) If no angle is obtuse, what can you conclude about the relative sizes of A, B and C? (iii) If A is obtuse, use the identity sin (180 ) sin  to explain why sin A is larger than sin B and sinC . Hence prove that A > B > C. 9. A ruined tower is fenced off for safety reasons. To find the height of the tower, Rachael stands at a point A and measures the angle of elevation as 18 . She then walks 20 metres directly towards the base of the tower to point B where the angle of elevation is 31 . By finding ATB , or otherwise, calculate the height, h, of the tower.

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 4 of 5 10. The following diagram shows a circle centred at (3, 4) with radius 1 unit , and a straight line that passes through the origin O. Given that th e straight line intersects the circle at least once, find the smallest and largest acute angle made by the line with the positive x-axis, giving your answers in radians correct to 3 decimal places. Challenging Questions 1. Prove that 11sin cos 2xx   for 1 1.x   2. Solve sin cos 1,where 0 2 .x x x    

National Junior College Mathematics Department 2016 Revision: Trigonometry Page 5 of 5 Answers to Trigonometry Tutorial Basic Mastery Questions 2. (a) 1 (b) 2 (c) 3 3. 2 2 4. (b)(i) 1 x (ii) 21 x (iii) 21 x x  5. 41, 3 6. 2sin 3x   7. 5or33x  8. 5 11or66x  9. 3 7 11 15, , or8 8 8 8x     10. (a) 23.6 (b) 16.0 cm (c) 6.01 cm (d) 89.6 Practice Questions 1. 33 63 63,,65 65 65 3. 50, , 2 , , or 33x  4. 37 or44x  5. XY 15.7cm, YZ 13.5cm 6. 32.2 7. (b) 21cos 22 4 2   (c)   2 2 2 30 f ( ) 4  8. (i) sin sin sinA B C 9. 14.2 metres 10. 0.726 rad, 1.129 rad Challenging Questions 2. 0, or 22x  