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 bounded by the curve with equation y = sec x, the x-axis and the lines x = and x = -, giving your answer to two decimal places.
(4)
(Total 8 marks)
3. Find, giving your answers to two decimal places, the values of w, x, y and z for which
(a) e-w = 4,
(2)
(b) arctan x = 1,
(2)
(c) ln (y + 1) – ln y = 0.85
(4)
(d) cos z + sin z = , -p < z < p.
(5)
(Total 13 marks)
4. (a) Using the formulae sin (A ± B) = sin A cos B ± cos A sin B, cos (A ± B) = cos A cos B sin A sin B, show that
(i) sin (A + B) – sin (A – B) = 2 cos A sin B,
(2)
(ii) cos (A – B) – cos (A + B) = 2 sin A sin B.
(2)
(b) Use the above results to show that = cot A.
(3)
Using the result of part (b) and the exact values of sin 60° and cos 60°, (c) find an exact value for cot 75° in its simplest form.
(4)
(Total 11 marks)
5. In a particular circuit the current, I amperes, is given by
I = 4 sin q – 3 cos q, q > 0, where q is an angle related to the voltage. Given that I = R sin (q - a), where R > 0 and 0 £ a < 360°, (a) find the value of R, and the value of a to 1 decimal place.
(4)
(b) Hence solve the equation 4 sin q – 3 cos q = 3 to find the values of q between
0 and 360°.
(5)
(c) Write down
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 bounded by the curve with equation y = sec x, the x-axis and the lines x = and x = -, giving your answer to two decimal places.
(4)
(Total 8 marks)
3. Find, giving your answers to two decimal places, the values of w, x, y and z for which
(a) e-w = 4,
(2)
(b) arctan x = 1,
(2)
(c) ln (y + 1) – ln y = 0.85
(4)
(d) cos z + sin z = , -p < z < p.
(5)
(Total 13 marks)
4. (a) Using the formulae sin (A ± B) = sin A cos B ± cos A sin B, cos (A ± B) = cos A cos B sin A sin B, show that
(i) sin (A + B) – sin (A – B) = 2 cos A sin B,
(2)
(ii) cos (A – B) – cos (A + B) = 2 sin A sin B.
(2)
(b) Use the above results to show that = cot A.
(3)
Using the result of part (b) and the exact values of sin 60° and cos 60°, (c) find an exact value for cot 75° in its simplest form.
(4)
(Total 11 marks)
5. In a particular circuit the current, I amperes, is given by
I = 4 sin q – 3 cos q, q > 0, where q is an angle related to the voltage. Given that I = R sin (q - a), where R > 0 and 0 £ a < 360°, (a) find the value of R, and the value of a to 1 decimal place.
(4)
(b) Hence solve the equation 4 sin q – 3 cos q = 3 to find the values of q between
0 and 360°.
(5)
(c) Write down