Download Questions - NLCS Maths Department

C3
1
Worksheet F
TRIGONOMETRY
Find all values of x in the interval 0 ≤ x ≤ 360° for which
tan2 x − sec x = 1.
2
(5)
a Express 2 cos x° + 5 sin x° in the form R cos (x − α)°, where R > 0 and 0 < α < 90.
(3)
b Solve the equation
2 cos x° + 5 cos x° = 3,
for values of x in the interval 0 ≤ x ≤ 360, giving your answers to 1 decimal place.
3
For values of θ in the interval 0 ≤ θ ≤ 360°, solve the equation
2 sin (θ + 30°) = sin (θ − 30°).
4
(3)
(6)
a Solve the equation
π − 6 tan−1 2x = 0,
giving your answer in the form k 3 .
(3)
b Find the values of x in the interval 0 ≤ x ≤ 360° for which
2 sin 2x = 3 cos x,
giving your answers to an appropriate degree of accuracy.
5
(5)
a Prove the identity
(1 − sin x)(sec x + tan x) ≡ cos x,
x≠
(2n + 1)π
,
2
n∈ .
(4)
b Find the values of y in the interval 0 ≤ y ≤ π for which
2 sec2 2y + tan2 2y = 3,
giving your answers in terms of π.
6
(5)
a Express 4 sin x° − cos x° in the form R sin (x − α)°, where R > 0 and 0 < α < 90.
(3)
b Show that the equation
2 cosec x° − cot x° + 4 = 0
(I)
can be written in the form
4 sin x° − cos x° + 2 = 0.
(2)
c Using your answers to parts a and b, solve equation (I) for x in the interval 0 ≤ x ≤ 360. (3)
7
a Prove the identity
cosec θ − sin θ ≡ cos θ cot θ,
θ ≠ nπ, n ∈ .
(3)
b Find the values of x in the interval 0 ≤ x ≤ 2π for which
2 sec x + tan x = 2 cos x,
giving your answers in terms of π.
8
(6)
a Sketch on the same diagram the curves y = 3 sin x° and y = 1 + cosec x° for x in the
interval −180 ≤ x ≤ 180.
(3)
b Find the x-coordinate of each point where the curves intersect in this interval, giving
your answers correct to 1 decimal place.
(5)
 Solomon Press
C3
TRIGONOMETRY
Worksheet F continued
9
a Prove the identity
cot 2x + cosec 2x ≡ cot x,
x≠
nπ
2
, n∈ .
(4)
b Hence, find to 2 decimal places the values of x in the interval 0 ≤ x ≤ 2π, such that
cot 2x + cosec 2x = 6 − cot2 x.
10
(5)
a Prove that for all real values of x
cos (x + 30)° + sin x° ≡ cos (x − 30)°.
(4)
b Hence, find the exact value of cos 75° − cos 15°, giving your answer in the form k 2 . (2)
c Solve the equation
3 cos (x + 30)° + sin x° = 3 cos (x − 30)° + 1,
for x in the interval −180 ≤ x ≤ 180.
11
(4)
y
(60, 5)
y = f(x)
(240, 1)
O
x
The diagram shows the curve y = f(x) where
f(x) ≡ a + b sin x° + c cos x°, x ∈ , 0 ≤ x ≤ 360,
The curve has turning points with coordinates (60, 5) and (240, 1) as shown.
a State, with a reason, the value of the constant a.
(2)
b Find the values of k and α, where k > 0 and 0 < α < 90, such that
f(x) = a + k sin (x + α)°.
(3)
c Hence, or otherwise, find the exact values of the constants b and c.
12
a Prove the identity
1 − cos x
x
≡ tan2 ,
2
1 + cos x
(2)
x ≠ (2n + 1)π, n ∈ .
(3)
b Use the identity in part a to
i
find the value of tan2
π
12
in the form a + b 3 , where a and b are integers,
ii solve the equation
1 − cos x
x
= 1 − sec ,
2
1 + cos x
for x in the interval 0 ≤ x ≤ 2π, giving your answers in terms of π.
13
a Express 2 sin x − 3 cos x in the form R sin (x − α), where R > 0 and 0 < α <
(8)
π
2
.
b State the minimum value of 2 sin x − 3 cos x and the smallest positive value of x for
which this minimum occurs.
(3)
(2)
c Solve the equation
2 sin 2x − 3 cos 2x + 1 = 0,
for x in the interval 0 ≤ x ≤ π, giving your answers to 2 decimal places.
 Solomon Press
(4)