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SKILLS
19.3
Trigonometry 2
Exercise 19.3S
1
You can use trigonometric ratios to find angles as well as sides.
sin θ =
O
sin θ = H
Opposite side
Adjacent side
Opposite side
cos θ =
tan θ =
Hypotenuse
Hypotenuse
Adjacent side
cos θ =
H
O
EXAMPLE
A
Find the angles marked by letters.
a
b
c 6 mm
3 cm
8 cm
x
20 mm
z
tan θ =
You can subtract
from 180° to find
the third angle if
you need it.
A
H
O
A
2
3
4.9 m
y
2.5 m
Identify which sides you know.
a A = 3, H = 8 b O = 6, A = 20 c O = 2.5, H = 4.9
4
Choose the ratio, substitute values, then write as a decimal.
x = cos-1 0.375
y = tan-1 0.3
= 68.0º (1 dp) = 16.7º (1 dp) = 30.7º (1 dp)
tan C = 0.43
dcos D = 0.37
e
sin E = 0.892
ftan F = 1.645
Calculate the size of each angle.
37
2
a tan P = bsin Q =
5
50
120
21
c cos R = dsin S =
25
150
198
7.8
e tan T =
fcos U =
435
5.4
Find the angles marked by letters.
c
EXAMPLE
9 cm
P
Q
8 cm
R
p. 52
If you have time, use the
other trigonometric ratios
to check your answers.
Use Pythagoras for the third side and
trigonometry for an angle.
8
PR2 = 92 – 82sin P = 9 = 0.8888…
= 81 – 64 ∠P = sin-1 0.8888…
= 17 = 62.73…
PR = !17 = 4.123… = 62.7° (3 sf)
= 4.1 cm (1 dp)
It is better to use
the sides you were
given rather
thanonthe
Carry
one you have
the found.
working
15 km
on your
calculator.
3m
3m
d
9m
10 km
9
d
a
A
f
b
3.6 m
D
F
12 m
8 cm
13 m
B
c
6 cm
E
C
G
d
J
40° 18 mm
34 m
L
72°
AQA_GCSE_Maths_SB_Sample02.indd 408-409
Pythagoras and trigonometry
R
42 cm
58 cm
O
Q
In each part, find the other side and angles
of triangle RST.
a
RS = 7 cm, RT = 10 cm and
angle R = 90°
b
RS = 36 m, RT = 54 m and angle S = 90°
c
RS = 3.5 km, TS = 1.67 km and
angle T = 90°
PQRS is a rectangle.
S
Calculate
a
QS
b
angle PQS
c
angle PSQ
R
4 cm
P
The diagonals of kite KLMN
intersect at X.
KX = 15 cm,
XM = 27 cm and K
LN = 24 cm.
Calculate
all the angles of
triangle PQR P
b the distance
from P to QR.
7 cm
Q
L
M
X
N
R
7 cm
4 cm
7 cm
*10 A rhombus has sides of length 10 cm.
Q
The length of the longest diagonal is 17 cm.
Find
the angles of the rhombus
b the length of the shortest diagonal.
11 The 20% on this road sign
H
Geometry and measures
P
a
the shortest side and the largest angle
opposite the longest side.
408
f
a
6.4 m
56 mm
450 km
Calculate the sides and
angles of kite KLMN.
Calculate all the unknown sides and angles.
Subtract from 180° to find the third angle.
∠Q = 180° – 90° – 62.7° Angle sum of a triangle.
= 27.3° (1 dp) Check that the smallest angle is opposite
f
35
mm
e
5
8
6m
c
e
7
b
a
4 cm
In a right-angled triangle given two sides or one side and one angle
you can find all the other sides and angles using trigonometry and
Pythagoras theorem.
Find the unknown sides
and angles in triangle PQR.
6
c
b
N
bsin B = 0.6
eM
15°
cos A = 0.5
10 cm
Don't round yet,
-1
carry
on working
z = sin 0.5102…
on your calculator.
5
5
8
120
290
a
a
2.5
6
3
cos x = 8 = 0.375 tan y = 20 = 0.3 sin z =
= 0.5102…
4.9
Use the inverse trigonometric function.
Write these fractions as decimals
9
3
a
b
c
4
10
6.3
16
d
e
f
2.7
25
Find the size of each angle.
I
K
1131
means the hill down up by
20 metres for every 100 metres
in the horizontal direction.
Find the angle
between the road
and the horizontal.
b How far along the road do you travel as
it falls by 20 metres?
a
SEARCH
409
10/10/14 12:14 PM