Trigonometry (Addition,Double Angle & R Formulae) - Edexcel Past Exam Questions. cos 2A º 1 2 sin 2 A. (2)

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1 Trigonometry (Addition,Double Angle & R Formulae) - Edexcel Past Exam Questions. (a) Using the identity cos (A + B) º cos A cos B sin A sin B, rove that cos A º sin A. () (b) Show that sin q 3 cos q 3 sin q + 3 º sin q (4 cos q + 6 sin q 3). (4) (c) Exress 4 cos q + 6 sin q in the form R sin (q + a ), where R > 0 and 0 < a <. (d) Hence, for 0 q <, solve (4) sin q = 3(cos q + sin q ), giving your answers in radians to 3 significant figures, where aroriate. (5) June 05 Q5. f(x) = cos x 4 sin x. Given that f(x) = R cos (x + a), where R ³ 0 and 0 a 90, (a) find the value of R and the value of a. (4) (b) Hence solve the equation cos x 4 sin x = 7 for 0 x < 360, giving your answers to one decimal lace. (5) (c) (i) Write down the minimum value of cos x 4 sin x. () (ii) Find, to decimal laces, the smallest ositive value of x for which this minimum value occurs. () Jan 06 Q6

2 3. (a) Show that cosx (i) º cos x sin x, x ¹ (n ), n Î Z, () 4 cos x + sin x (ii) (cos x sin x) º cos x cos x sin x. (3) (b) Hence, or otherwise, show that the equation æ cosq ö cos q ç = è cosq + sinq ø can be written as (c) Solve, for 0 q <, sin q = cos q. (3) sin q = cos q, giving your answers in terms of. (4) Jan 06 Q (a) Given that cos A =, where 70 < A < 360, find the exact value of sin A. 4 æ ö æ ö (b) Show that cos çx + + cos çx - º cos x. (3) è 3 ø è 3 ø June 06 Q8(edited) (5) 5. (a) By writing sin 3q as sin (q + q ), show that sin 3q = 3 sin q 4 sin 3 q. (5) Ö3 (b) Given that sin q =, find the exact value of sin 3q. 4 () Jan 07 Q

3 6. Figure y x Figure shows an oscilloscoe screen. The curve on the screen satisfies the equation y = Ö3 cos x + sin x. (a) Exress the equation of the curve in the form y = R sin (x + a ), where R and a are constants, R > 0 and 0 < a <. (4) (b) Find the values of x, 0 x <, for which y =. (4) Jan 07 Q5 7. (a) Exress 3 sin x + cos x in the form R sin (x + α) where R > 0 and 0 < α <. (4) (b) Hence find the greatest value of (3 sin x + cos x) 4. () (c) Solve, for 0 < x < π, the equation 3 sin x + cos x =, giving your answers to 3 decimal laces. (5) June 07 Q6

4 8. (a) Prove that sinq cosq + = cosec q, q ¹ 90n. (4) cosq sinq (b) Sketch the grah of y = cosec θ for 0 < θ < 360. () (c) Solve, for 0 < θ < 360, the equation sinq cosq + = 3 cosq sinq giving your answers to decimal lace. (6) June 07 Q7 9. (a) Use the double angle formulae and the identity cos(a + B) cosa cosb sina sinb to obtain an exression for cos 3x in terms of owers of cos x only. (4) (b) (i) Prove that cos x + sin x + º sec x, x (n + ). (4) + sin x cos x (ii) Hence find, for 0 < x < π, all the solutions of cos x + sin x + = 4. (3) + sin x cos x Jan 08 Q6

5 0. f(x) = 5 cos x + sin x. Given that f(x) = R cos (x α), where R > 0 and 0 < α <, (a) find the value of R and the value of α to 3 decimal laces. (4) (b) Hence solve the equation 5 cos x + sin x = 6 for 0 x < π. (5) (c) (i) Write down the maximum value of 5 cos x + sin x. () (ii) Find the smallest ositive value of x for which this maximum value occurs. () June 08 Q. (a) (i) By writing 3θ = (θ + θ), show that sin 3θ = 3 sin θ 4 sin 3 θ. (4) (ii) Hence, or otherwise, for 0 < θ <, solve 3 8 sin 3 θ 6 sin θ + = 0. Give your answers in terms of π. (5) (b) Using sin (θ a) = sin θ cos a cos θ sin a, or otherwise, show that sin 5 = (Ö6 Ö). (4) 4 Jan 09 Q6

6 . (a) Use the identity cos (A + B) = cos A cos B sin A sin B, to show that The curves C and C have equations cos A = sin A () C: y = 3 sin x C: y = 4 sin x cos x (b) Show that the x-coordinates of the oints where C and C intersect satisfy the equation 4 cos x + 3 sin x = (3) (c) Exress 4cos x + 3 sin x in the form R cos (x α), where R > 0 and 0 < α < 90, giving the value of α to decimal laces. (3) (d) Hence find, for 0 x < 80, all the solutions of 4 cos x + 3 sin x =, giving your answers to decimal lace. (4) June 09 Q6 3. (a) Exress 5 cos x 3 sin x in the form R cos(x + α), where R > 0 and 0 < α <. (4) (b) Hence, or otherwise, solve the equation 5 cos x 3 sin x = 4 for 0 x <, giving your answers to decimal laces. (5) Jan 0 Q3 4. Solve for 0 x 80. cosec x cot x = (7) Jan 0 Q8

7 5. (a) Show that sin θ = tan θ. () + cos θ (b) Hence find, for 80 θ < 80, all the solutions of sin θ + cos θ =. Give your answers to decimal lace. (3) June 0 Q 6. (a) Exress 7 cos x 4 sin x in the form R cos (x + a) where R > 0 and 0 < a <. Give the value of a to 3 decimal laces. (3) (b) Hence write down the minimum value of 7 cos x 4 sin x. () (c) Solve, for 0 x <, the equation 7 cos x 4 sin x = 0, giving your answers to decimal laces. (5) Jan Q 7. Find all the solutions of cos q = sin q in the interval 0 q < 360. (6) Jan Q3 8. (a) Prove that (b) Hence, or otherwise, cosq - = tan q, q ¹ 90n, n Î Z. (4) sin q sin q (i) show that tan 5 = Ö3, (3) (ii) solve, for 0 < x < 360, cosec 4x cot 4x =. (5) June Q6

8 9. (a) Exress cos 3x 3 sin 3x in the form R cos (3x + a), where R and a are constants, R > 0 and 0 < a <. Give your answers to 3 significant figures. (4) June Q8(edited)