The Sine Rule for Solving Triangles
26 February 2025
The sine rule
LO: Using the sine rule to solve non-right-
angled triangles
You can use trigonometry to solve triangles that are
not
right-angled.
Look at the triangle
ABC
.
a
b
c
D
h
perpendicular to
AC
.
In the right-angled triangle
ABD
.
This gives
h
=
c
sin
A
In the right-angled triangle
BCD
.
This gives
h
=
a
sin
C
Equate the values of
h
to give
=
a
sin
C
c
sin
A
The sine rule
Rearranging this equation
gives
The altitude (height),
h
of the
triangle is
BD
The ratios of the sine of each
angle to the length of the
opposite side are equal
You can use trigonometry to solve triangles that are
not
right-angled.
Look at the triangle
ABC
.
a
b
c
E
h
perpendicular to
AB
.
In the right-angled triangle
ACE
.
This gives
h
=
b
sin
A
In the right-angled triangle
BCE
.
This gives
h
=
a
sin
B
Equate the values of h to give
=
a
sin
B
b
sin
A
Rearranging this equation
gives
Now draw the altitude (height),
h
from
C
to
E
The ratios of the sine of each
angle to the length of the
opposite side are equal
The sine rule
b
c
a
or
The sine rule is a method to calculate sides and angles of any
triangle. It is useful for finding the length of one side when
all the angles and one other side are known, or finding an
angle when two sides and one other angle are known
To find an angle, use the
sine rule written as
To find a side, use the
sine rule written as
Where
The sine rule
Using the sine rule to find side lengths
If we are given two angles in a triangle and the length of a
side opposite one of the angles, we can use the sine rule to
find the length of the side opposite the other angle.
For example:
Find the length of side
a
Using the sine rule:
a
=
9.82
cm
(to 3 s.f.)
Using the sine rule to find side lengths
Find the length of side
x,
rounded to 2 decimal places
Using the sine rule:
x
≈
25.26
m
(to 2 dp)
If we are given two
side lengths
in a triangle and the
angle
opposite one of the
given sides
, we can use the sine rule to
find the
angle
opposite the other
given side
. For example:
Find the
angle at
A
Using the sine rule:
A
=
41.7
°
(to 3 s.f.)
Using the sine rule to find angles
Using the sine rule:
9
cm
4
5
°
R
7
cm
Q
P
Using the sine rule to find angles
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info@mathssupport.org
For more resources visit our website
The Sine Rule in trigonometry allows for solving non-right-angled triangles by relating the ratios of the sine of angles to the lengths of corresponding sides. This method is essential in finding missing angles or side lengths in triangles, providing a systematic approach for trigonometric calculations. Explore how to apply the Sine Rule to solve triangles efficiently and accurately with practical examples and step-by-step explanations.
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Presentation Transcript
26 February 2025 The sine rule LO: Using the sine rule to solve non-right- angled triangles www.mathssupport.org
The sine rule You can use trigonometry to solve triangles that are not right-angled. Look at the triangle ABC. perpendicular to AC. In the right-angled triangle ABD. ? ? This gives h = c sin A In the right-angled triangle BCD. ? ? This gives h = a sin C Equate the values of h to give = a sin C c sin A The altitude (height), h of the B triangle is BD a c h ???? = C A b D Rearranging this equation gives sin? ? The ratios of the sine of each angle to the length of the opposite side are equal ???? = =sin? ? www.mathssupport.org
sin? ? =sin? =sin? The sine rule ? ? You can use trigonometry to solve triangles that are not right-angled. Look at the triangle ABC. perpendicular to AB. In the right-angled triangle ACE. ? ? This gives h = b sin A In the right-angled triangle BCE. ? ? This gives h = a sin B Equate the values of h to give = a sin B b sin A Now draw the altitude (height), h B from C to E E a c h ???? = C A b Rearranging this equation gives sin? ? The ratios of the sine of each angle to the length of the opposite side are equal ???? = =sin? ? www.mathssupport.org
The sine rule The sine rule is a method to calculate sides and angles of any triangle. It is useful for finding the length of one side when all the angles and one other side are known, or finding an angle when two sides and one other angle are known To find a side, use the sine rule written as For any triangle ABC: C b c a = = b a sin B sin C sin A or A B To find an angle, use the sine rule written as c Where ais the length of the side opposite ? bis the length of the side opposite ? cis the length of the side opposite ? sin A a sin B b sin C c = = www.mathssupport.org
Using the sine rule to find side lengths If we are given two angles in a triangle and the length of a side opposite one of the angles, we can use the sine rule to find the length of the side opposite the other angle. For example: Find the length of side a B Using the sine rule: 39 7 a a = sin 39 sin 118 118 7 sin 118 sin 39 a = C A 7 cm a = 9.82 cm (to 3 s.f.) www.mathssupport.org
Using the sine rule to find side lengths Find the length of side x, rounded to 2 decimal places Using the sine rule: 18 x A 18 m sin 113 = sin 41 C 113 18 sin 113 sin 41 41 x = x B x 25.26 m (to 2 dp) www.mathssupport.org
Using the sine rule to find angles If we are given two side lengths in a triangle and the angle opposite one of the given sides, we can use the sine rule to find the angle opposite the other given side. For example: Find the angle at A Using the sine rule: sin A 9.4 C sin 98 = 14 14 cm 9.4 cm 9.4 sin 98 14 sin A = 9.4 sin 98 14 98 A B sin 1 A = A = 41.7 (to 3 s.f.) www.mathssupport.org
Using the sine rule to find angles The following diagram shows PQR, find ? ??. P Using the sine rule: sin ? ?? sin 45 7 = 7 cm 9 9 sin 45 7 sin ? ?? = 45 Q R 9 cm 9 sin 45 7 sin 1 ? ?? = Finding angle ? ?? ? ??= 180 45 65.4 ? ??= 65.4 (to 3 s.f.) ? ??= 69.6 www.mathssupport.org
Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org www.mathssupport.org
