Trigonometry Summer Work
Trigonometry is a vital part of year 13 maths so you need to be confident with
it. If you struggle with any part of this work you need to do some revision over
summer to improve your trigonometry skills (use your notes, Dr Frost Maths,
the textbook and revision websites)
Please answer all 10 questions on separate paper and bring it with you to the
first lesson back. If you are not printing this document please write out the
questions. You need to label each question and set your working out clearly.
You must attempt all questions.
Key Points:
• tan 𝜃𝜃 =
sin 𝜃𝜃
cos 𝜃𝜃
• 𝑠𝑠𝑠𝑠𝑠𝑠2 𝑥𝑥 + 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑥𝑥 = 1
• 180 degrees = π radians
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1. Find all the solutions, in the interval 0 ⩽ x < 2π, of the equation
2 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑥𝑥 + 1 = 5 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
giving each solution in terms of π
(6)
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2. Solve, for 0 ⩽ x < 180◦,
cos(3𝑥𝑥 − 10𝑜𝑜 ) = −0.4
giving your answers to 1 decimal place
(7)
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𝜋𝜋
3. a) Sketch for 0 ⩽ x ⩽ 2π, the graph of 𝑦𝑦 = 𝑠𝑠𝑠𝑠𝑠𝑠 �𝑥𝑥 + �
6
(2)
b) Write down the exact coordinates of the points where the graph meets the
coordinate axes
(3)
c) Solve for 0 ⩽ x ⩽ 2π, the equation
𝜋𝜋
𝑠𝑠𝑠𝑠𝑠𝑠 �𝑥𝑥 + � = 0.65
6
giving your answers to 2 decimal places
(5)
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4. a) Show that the equation
can be written as
4 𝑠𝑠𝑖𝑖𝑖𝑖2 𝑥𝑥 + 9𝑐𝑐𝑐𝑐𝑐𝑐𝑥𝑥 − 6 = 0
4 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑥𝑥 − 9𝑐𝑐𝑐𝑐𝑐𝑐𝑥𝑥 + 2 = 0
(2)
b) Hence solve for 0 ⩽ x < 720◦,
4𝑠𝑠𝑖𝑖𝑖𝑖2 𝑥𝑥 + 9𝑐𝑐𝑐𝑐𝑐𝑐𝑥𝑥 − 6 = 0
giving your answers to 1 decimal place.
(6)
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5. (i) Solve for -180◦ ⩽ x < 180◦
(1 + 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)(5𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 2) = 0𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
(ii) Solve for 0 ⩽ x < 360◦,
4 𝑠𝑠𝑠𝑠𝑠𝑠𝑥𝑥 = 3 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
(4)
(6)
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6. a) Solve for 0 ⩽ x < 360◦, giving your answers in degrees to 1 decimal place
3 𝑠𝑠𝑠𝑠𝑠𝑠(𝑥𝑥 + 45) = 2𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
(4)
b) Find, for 0 ⩽ x < 2π, all the solutions of
2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝑥𝑥 + 2 = 7 cos 𝑥𝑥
giving your answers in radians
(6)
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7.
Giving your answers in terms of π, solve the equation
3 tan2 x – 1 = 0
For x in the interval -π ⩽ x ⩽ π.
(6)
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8.
Find all the solutions, in the interval 0 ⩽ x ⩽ 360°, to the equation
8 – 7 cos x = 6 sin2 x
giving solutions to 1 decimal place where appropriate.
(6)
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9.
Find, to 1 decimal place, the values of θ in the interval 0 ⩽ θ ⩽ 180° for
which
4√3 sin (3θ + 20°) = 4 cos (3θ + 20°).
(6)
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10. (a) Calculate the value of −2 tan (−120°).
(1)
(b) On the same set of axes sketch the graphs of y = 2 sin (x − 60°) and
y = −2 tan x, in the interval −180° ⩽ x ⩽ 180°, showing the coordinates of
points of intersection with the coordinate axes in exact form.
(7)
(c) Explain how you can use the graph to identify solutions to the equations
y = 2 sin (x − 60°) + 2 tan x = 0 in the interval −180° ⩽ x ⩽ 180°.
(1)
(d) Write down the number of solutions of the equation
y = 2 sin (x − 60°) + 2 tan x = 0 in the interval −180° ⩽ x ⩽ 180°.
(1)
(Total 10 marks)
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