C3 Trig Questions

Archbishop Tenison's School C3 Trigonometry 340 min 283 marks 1. (a) Prove that for all values of x, sin x + sin (60° - x) º sin (60° + x). (4) (b) Given that sin 84° - sin 36° = sin a°, deduce the exact value of the acute angle a. (2) (c) Solve the equation 4 sin 2x + sin (60° - 2x) = sin (60° + 2x) – 1 for values of x in the interval 0 £ x < 360°, giving your answers to one decimal place. (5) (Total 11 marks) 2. On separate diagrams, sketch the curves with equations (a) y = arcsin x, -1 £ x £ 1, (b) y = sec x, - £ x £ , stating the coordinates of the end points of your curves in each case. (4) Use the trapezium rule with five equally spaced ordinates to estimate the area of the region …show more content…

(3) (c) Solve, for 0 £ θ £ 2π, sin 2θ = cos 2θ, giving your answers in terms of π. (4) (Total 12 marks) 18. (a) Differentiate with respect to x (i) x2e3x+2, (4) (ii) (4) (b) Given that x = 4 sin(2y + 6), find in terms of x. (5) (Total 13 marks) 19. f(x) = 12 cos x – 4 sin x. Given that f(x) = R cos(x + α), where R ³ 0 and 0 £ α £ 90°, (a) find the value of R and the value of α. (4) (b) Hence solve the equation 12 cos x – 4 sin x = 7 for 0 £ x £ 360°, giving your answers to one decimal place. (5) (c) (i) Write down the minimum value of 12 cos x – 4 sin x. (1) (ii) Find, to 2 decimal places, the smallest positive value of x for which this minimum value occurs. (2) (Total 12 marks) 20. (a) By writing sin 3θ as sin (2θ + θ), show that sin 3θ = 3sinθ – 4sin3θ. (5) (b) Given that sinθ = , find the exact value of sin 3θ. (2) (Total 7 marks) 21. The diagram above shows an oscilloscope screen. The curve shown on the screen satisfies the equation (a) Express the equation of the curve in the form y = Rsin(x + α), where R and α are constants, R > 0 and 0 < α < (4) (b) Find the values of x, 0 ≤ x < 2p, for which y = 1. (4) (Total 8 marks) 22. (i) Prove that sec2x – cosec2 x ≡ tan2x – cot2x. (3) (ii) Given that y = arccos x, –1 ≤ x ≤ 1, and 0 ≤ y ≤ π (a) express arcsin x in terms of y.