SPM Past Year Questions : AM Form 5 Chapter 5 Trigonometric Functions

Size: px

Start display at page:

Download "SPM Past Year Questions : AM Form 5 Chapter 5 Trigonometric Functions"

Ferdinand Pope
6 years ago
Views:

1 SPM 1993 SPM PAST YEAR QUESTIONS ADDITIONAL MATHEMATICS FORM 5 CHAPTER 5 : TRIGONOMETRIC FUNCTIONS 1. Solve the equation sec x = 3 tan x for 0 x 360. [5 marks]. Given that tan θ = 1, without using a calculator, find the value of 3 (a) tan θ, (b) tan 135 θ. [5 marks] 3.(a) If g= cos 15 and h=sin 35, state in terms of g and/or h, (i) cos 50 (ii) cos 70 (iii) cos 7 1 [7 marks] (b) Sketch the graph of y = sin x for 0 x π. State the suitable equation of straight line used to solve the equation x π sin x = 0 for 0 x π. Sketch the suitable straight line and state the number of solutions for the equation. 4.(a) Prove that cosec A + cot A = cot A. (b) Solve the following equation for 0 x 360. (i) sin x = tan x, (ii) 3 cos x sin x = 1. [8 marks] SPM (a) Derive the identity cos θ = cos θ 1. (b) Given that θ is an acute angle and sin θ = p. Express each of the following in terms of p. (i) tan θ, (ii) cos ( θ), (iii) cos θ 6. Sketch the graph for f x = sin x + 1 in the domain of 0 x (a) Solve each of the following equation for 0 θ (i) tan θ + cosec θ 3 = 0, (ii) 3 cos θ + sin θ = 0. [7 marks] (b) Find the values of A and B which satisfy the equation sin(a+b) = 0.8 and sin(a - B) = 0.4 with the condition 0 A + B 90 and 0 A B 90. SPM Solve each of the following equation for 90 β 70. (a) tan β = 1. (b) 3 sin β cos β = 0. 9.(a) Prove the identity (cos θ + 1) tan θ = sin θ. Compiled by Miss Page 1

2 (b) Using the scale of cm to π rad on the x-axis and 1 cm to 1 unit on the y-axis, draw the graph 6 of y = 5 tan x for 0 x 3π. On the same axes, draw the suitable straight line to solve the equation 30 tan x = 6x 5π for the domain of 0 x 3π. Hence, find the value of x in rad. 10.Given tan A 3 tan B = 1 and tan(a+b) = with A and B are acute angles. Find the values of A and B which satisfy both equations. [5 marks] SPM On the same axes, sketch the graph y = 1 + sin x and y = cos x for 0 x π. Hence, state the number of solutions for the equation cos x = 1 + sin x. [5 marks] 1.(a) Solve each of the following equations for all the angles between 0 and 360. (i) 4 cos x = 3 cos ecx 4 (ii) 5 tan y cot y = sec y [8 marks] (b)given that tan θ = n where θ is an acute angle. Express cos(180 θ) in terms of n. SPM (a) Solve for 0 x π the equation of 4 sin x π cos x π = 1. (b) Given that tan y = 5 1 cos y. 14.By sketching the graph of y = + cos x for 0 x π, find the range of values of k such that cos x = k does not have any root. 15.(a) Show that sin θ+sin θ 1+cos θ + cos θ = tan θ. (b) Solve for 0 α 360 the equation of cos α 60 = sin α. SPM Given sin θ = k with θ, an acute angle. Find (a) sin θ in terms of k, (b) the positive value of k if cos θ = k. 17.(a) Prove the identity tan θ cot θ = sec θ cos ec θ. (b) Prove that the equation sin x + k sin x + 3k = 9 has real roots for all the values of k. (c)using the scale of cm to 0.5 unit on the x-axis and the y-axis, draw the graph of y = cos π x for 0 x 3. From your graph, estimate the values of x which satisfy the equation cos π x + 14=0 in the domain of 0 x 3. [5 marks] Compiled by Miss Page

3 18.Given sin x = 3 5 and x is an acute angle. Without using a calculator, find the value of (a) cos π x, (b) cos ecx + cos x, (c) sec x. [5 marks] SPM (a) Solve cos x sin x = 1 for 0 x (b) Given cos x = 3 5 and sin y = with x in the second quadrant and y in the third quadrant Without using a calculator, find the value of sin x y. 0.(a)Solve the equation 5 cos θ = 5 3 sin θ for 0 x 90. (b)(i) Sketch the graph of y = 3 cos x for 0 x π. (ii) Hence, with the aid of the graph at (b)(i), determine the value of k such that cos x = 3k has only two solutions for 0 x π. 1. The given diagram shows the graph of a function f x. State the function. SPM Given cot θ = (p 1) (a) if sin θ = cos θ, (b) if sin θ = 5 3. for π θ π. Find the value of p 3. Solve 6 sin θ π 3 sin θ π 3 = 0 for 0 θ 360. [5 marks] 4. Given that cos x y = 1 and cos x cos y = 3, find the value of 4 (a) sin x sin y, (b) cos x + y. SPM (a) Prove that tan θ sin θ = tan θ sin θ. (b)without using a calculator, solve the equation sin x = cos( 5 ) for 0 x (a)Prove that tan θ + 1 tan θ = sin θ. Compiled by Miss Page 3

4 (b) Given that sin A = 5 3 value of and angle A is an obtuse angle. Without using a calculator, find the (i) cos A (ii) sin A (c) Sketch the graph of y = cos x for 0 x π. On the same axes, sketch a suitable straight line to solve the equation x π cos x = 0. Hence, state the number of solutions for the equation for 0 x π. 7.Given that cos 50 = p and sin 5 = q. Without using mathematical tables or a calculator, (a) prove that p + q = 1, (b) express sin 75 in terms of p and q. [5 marks] SPM 00 8.Solve the equation sin x = cos x for 0 x Given that tan θ = p and sin θ = p. Without using mathematical tables or a calculator, find the value of tan 15. SPM p 30. Given that tan θ = t, 0 < θ < 90, express in terms of t: (a) cot θ (b) sin(90 θ) 31. Solve the equation 6 sec A 13 tan A = 0, 0 A (a) Prove that tan θ + cot θ = cos ec θ. (b)(i) Sketch the graph of y = cos 3 x for 0 x π. (ii) Find the suitable equation of straight line used to solve the equation cos 3 x = 3 4π x 1. Hence, using the same axes, sketch the straight line and state the number of solutions to the equation cos 3 x = 3 x 1 for 0 x π. 4π SPM Solve the equation cos x sin x = sin x for 0 x (a) Sketch the graph of y = cos x for 0 x 180. (b) Hence, by sketching a suitable straight line within the same axes, find the number of solutions for the equation sin x = x for 0 x Compiled by Miss Page 4

5 SPM Solve the equation 3 cos x = 8 sin x 5 for 0 x (a) Prove that cosec x sin x cot x = cos x. (b)(i) Sketch the graph of y = cos x for 0 x π. (ii) Hence, by using the same axes, draw a suitable straight line to find the number of solution to the equation 3(cosec x sin x cot x) = x 1 for 0 x π. State the number of π solutions. SPM Solve the equation 15 sin x = sin x + 4 sin 30 for 0 x (a) Sketch the graph of y = cos x for 0 x π. (b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the equation π + cos x = 0 for 0 x π. State the number of solutions. x SPM Solve the equation cot x + cos x = 0 for 0 x (a) Sketch the graph of y = 3 cos x for 0 x π. (b) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation 3 cos x = x for 0 x π. State the number of solutions. π Compiled by Miss Page 5

Trigonometric Identities Exam Questions

Trigonometric Identities Exam Questions Name: ANSWERS January 01 January 017 Multiple Choice 1. Simplify the following expression: cos x 1 cot x a. sin x b. cos x c. cot x d. sec x. Identify a non-permissible