C3 Exam Workshop 2
Workbook
1.
(a) Express 7 cos x − 24 sin x in the form R cos (x + α) where R > 0 and 0 < α <
Give the value of α to 3 decimal places.
π
2
.
(3)
(b) Hence write down the minimum value of 7 cos x – 24 sin x.
(1)
(c) Solve, for 0 ≤ x < 2π, the equation
7 cos x − 24 sin x = 10,
giving your answers to 2 decimal places.
(5)
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2.
(a) Express 1.5 sin 2x + 2 cos 2x in the form R sin (2x + α), where R > 0 and
0 < α < 12 π , giving your values of R and α to 3 decimal places where appropriate.
(4)
(b) Express 3 sin x cos x + 4 cos2 x in the form a cos 2x + b sin 2x + c, where a, b and
c are constants to be found.
(2)
(c) Hence, using your answer to part (a), deduce the maximum value of 3 sin x cos x +
4 cos2 x.
(2)
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3.
(a) Prove that
1 − cos 2θ
≡ tan θ ,
sin 2θ
θ ≠
nπ
, n ∈ ℤ.
2
(3)
(b) Solve, giving exact answers in terms of π,
2(1 – cos 2θ ) = tan θ ,
0<θ <π.
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(6)
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4.
(a) Express sin x + √3 cos x in the form R sin (x + α), where R > 0 and 0 < α < 90°.
(4)
(b) Show that the equation sec x + √3 cosec x = 4 can be written in the form
sin x + √3 cos x = 2 sin 2x.
(3)
(c) Deduce from parts (a) and (b) that sec x + √3 cosec x = 4 can be written in the
form
sin 2x – sin (x + 60°) = 0.
(1)
X +Y
X −Y
, or otherwise,
sin
2
2
find the values of x in the interval 0 ≤ x ≤ 180°, for which sec x + √3 cosec x = 4.
(5)
(d) Hence, using the identity sin X – sin Y = 2 cos
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5.
(i) (a) Express (12 cos θ – 5 sin θ) in the form R cos (θ + α), where R > 0 and
0 < α < 90°.
(4)
(b) Hence solve the equation
12 cos θ – 5 sin θ = 4,
for 0 < θ < 90°, giving your answer to 1 decimal place.
(3)
(ii) Solve
8 cot θ – 3 tan θ = 2,
for 0 < θ < 90°, giving your answer to 1 decimal place.
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(5)
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6.
(i) Given that sin x =
3
, use an appropriate double angle formula to find the exact
5
value of sec 2x.
(4)
(ii) Prove that
cot 2x + cosec 2x ≡ cot x,
nπ
⎛
⎞
, n ∈ Z⎟ .
⎜x≠
2
⎝
⎠
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(4)
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7.
(a) Prove that
1 − tan 2 θ
≡ cos 2θ .
1 + tan 2 θ
(4)
(b) Hence, or otherwise, prove
tan2
π
8
= 3 – 2√2.
(5)
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8.
(a) Prove that for all values of x
2 tan x − sin 2x = 2 sin 2 x tan x .
(5)
(b) Hence, or otherwise, find the values of x in the interval 0 ≤ x ≤ 360º, for which
2 tan x − sin 2x = sin 2 x
giving your answers to an appropriate degree of accuracy.
(6)
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9.
(a) Using the half-angle formulae, or otherwise, prove that for all values of x
1 + cos x
x
≡ cot 2 .
1 − cos x
2
(5)
(b) Hence, or otherwise, find the values of x in the interval 0 ≤ x ≤ 2π for which
1 + cos x
x
= 6cosec − 10
1 − cos x
2
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(7)
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10.
(a) Prove that there are no real values of θ for which
cos 2θ + cosθ + 2 = 0 .
(4)
(b) Find the values of x in the interval 0 ≤ x ≤ 360º, for which
3sin x − 2 cos 2 x = 0
(5)
(c) Hence, find the values of y in the interval 0 ≤ y ≤ 180º, for which
3sec 2y − 2 cot 2y = 0
(4)
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11.
(a) Prove that for all values of x
(
)
cos 2 x − sin 2 2x ≡ cos 2 x 4 cos 2 x − 3 .
(5)
(b) Hence, or otherwise, find the values of x in the interval 0 ≤ x ≤ 2π for which
cos 2 x − sin 2 2x = 0
giving your answer in terms of π.
(6)
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12.
(a) Show that for all values of x, where x is measured in degrees,
cos ( x + 60° ) − 3 sin ( x − 60° ) ≡ 2 cos x − 3 sin x .
(5)
(b) Hence, find the values of x in the interval -180º ≤ x ≤ 180º, for which
cos ( x + 60º ) − 3 sin ( x − 60º ) = 0
(4)
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13.
(a) Using the identity cos (A + B) ≡ cos A cos B – sin A sin B, prove that
cos 2A ≡ 1 – 2 sin2 A.
(2)
(b) Show that
2 sin 2θ – 3 cos 2θ – 3 sin θ + 3 ≡ sin θ (4 cos θ + 6 sin θ – 3).
(4)
(c) Express 4 cos θ + 6 sin θ in the form R sin (θ + α ), where R > 0 and
0 < α < 12 π .
(4)
(d) Hence, for 0 ≤ θ < π, solve
2 sin 2θ = 3(cos 2θ + sin θ – 1),
giving your answers in radians to 3 significant figures, where appropriate.
(5)
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14.
(a) Use the double angle formulae and the identity
cos(A + B) ≡ cosA cosB − sinA sinB
to obtain an expression for cos 3x in terms of powers of cos x only.
(4)
(b) (i) Prove that
cos x
1+ sin x
+
≡ 2 sec x,
1 + sin x
cos x
x ≠ (2n + 1)
π
.
2
(4)
(ii) Hence find, for 0 < x < 2π, all the solutions of
cos x
1+ sin x
+
= 4.
1 + sin x
cos x
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(3)
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15.
a)
b)
Given that sin (θ + α ) = 2.5 sin θ , show that tan θ =
sin α
.
2.5 − cos α
(3)
Hence, solve the equation sin (θ + 45° ) = 2.5 sin θ , given 0° ≤ θ ≤ 360° .
(4)
END
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