ALEVELS P3
T1 TRIG RSIN/ RCOS
By expressing 8 sin θ − 6 cos θ in the form R sin(θ − α), solve the equation
1
8 sin θ − 6 cos θ = 7,
for 0◦ ≤ θ ≤ 360◦ .
[7]
9709/03/O/N/05
( i) Express 7 cos θ + 24 sin θ in the form R cos(θ − α ), where R > 0 and 0◦ < α < 90◦ , giving the
[3]
exact value of R and the value of α correct to 2 decimal places.
2
(ii) Hence solve the equation
7 cos θ + 24 sin θ = 15,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .
[4]
9709/03/M/J/06
( i) Express 5 sin x + 12 cos x in the form R sin(x + α ), where R > 0 and 0◦ < α < 90◦ , giving the
[3]
value of α correct to 2 decimal places.
3
(ii) Hence solve the equation
5 sin 2θ + 12 cos 2θ = 11,
giving all solutions in the interval 0◦ < θ < 180◦ .
[5]
9709/03/O/N/08
4
√
√
(i) Express ( 6) cos θ + ( 10) sin θ in the form R cos(θ − α ), where R > 0 and 0◦ < α < 90◦ . Give
the value of α correct to 2 decimal places.
[3]
(ii) Hence, in each of the following cases, find the smallest positive angle θ which satisfies the
equation
√
√
(a) ( 6) cos θ + ( 10) sin θ = −4,
[2]
√
√
(b) ( 6) cos 12 θ + ( 10) sin 12 θ = 3.
[4]
9709/33/O/N/10
5
(i) Express cos x + 3 sin x in the form R cos(x − α ), where R > 0 and 0◦ < α < 90◦ , giving the exact
value of R and the value of α correct to 2 decimal places.
[3]
(ii) Hence solve the equation cos 2θ + 3 sin 2θ = 2, for 0◦ < θ < 90◦ .
[5]
9709/31/O/N/11
6
(i) Express 8 cos θ + 15 sin θ in the form R cos(θ − α ), where R > 0 and 0◦ < α < 90◦ . Give the value
of α correct to 2 decimal places.
[3]
(ii) Hence solve the equation 8 cos θ + 15 sin θ = 12, giving all solutions in the interval 0◦ < θ < 360◦ .
[4]
9709/33/O/N/11
7
(i) Express 24 sin θ − 7 cos θ in the form R sin(θ − α ), where R > 0 and 0◦ < α < 90◦ . Give the value
of α correct to 2 decimal places.
[3]
(ii) Hence find the smallest positive value of θ satisfying the equation
24 sin θ − 7 cos θ = 17.
[2]
9709/33/O/N/12
8
(i) Given that sec 1 + 2 cosec 1 = 3 cosec 21, show that 2 sin 1 + 4 cos 1 = 3.
[3]
(iii) Hence solve the equation sec 1 + 2 cosec 1 = 3 cosec 21 for 0Å < 1 < 360Å.
[4]
(ii) Express 2 sin 1 + 4 cos 1 in the form R sin 1 + ! where R > 0 and 0Å < ! < 90Å, giving the value
of ! correct to 2 decimal places.
[3]
9709/33/O/N/13
9
(i) Express 3 sin 1 + 2 cos 1 in the form R sin 1 + !, where R > 0 and 0Å < ! < 90Å, stating the exact
[3]
value of R and giving the value of ! correct to 2 decimal places.
(ii) Hence solve the equation
for 0Å < 1 < 180Å.
3 sin 1 + 2 cos 1 = 1,
[3]
9709/32/M/J/15
10
(i) Express ï5 cos x + 2 sin x in the form R cos x − !, where R > 0 and 0Å < ! < 90Å, giving the
value of ! correct to 2 decimal places.
[3]
(ii) Hence solve the equation
for 0Å < x < 360Å.
11
ï5 cos 21 x + 2 sin 21 x = 1.2,
[3]
9709/33/M/J/16
(i) By first expanding 2 sin x − 30Å, express 2 sin x − 30Å − cos x in the form R sin x − !, where
R > 0 and 0Å < ! < 90Å. Give the exact value of R and the value of ! correct to 2 decimal places.
[5]
(ii) Hence solve the equation
for 0Å < x < 180Å.
2 sin x − 30Å − cos x = 1,
[3]
9709/31/M/J/17
12
(i) Show that the equation ï2 cosec x + cot x = ï3 can be expressed in the form R sin x − ! = ï2,
where R > 0 and 0Å < ! < 90Å.
[4]
(ii) Hence solve the equation ï2 cosec x + cot x = ï3, for 0Å < x < 180Å.
[4]
9709/31/O/N/18
13
(i) Express ï6 sin x + cos x in the form R sin x + !, where R > 0 and 0Å < ! < 90Å. State the exact
value of R and give ! correct to 3 decimal places.
[3]
(ii) Hence solve the equation ï6 sin 21 + cos 21 = 2, for 0Å < 1 < 180Å.
[4]
9709/32/O/N/19