Download Page Trigonometry can be used to calculate the side lengths and

Trigonometry — Sine Law and Cosine Law
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Trigonometry can be used to calculate the side lengths and angle measures
of triangles. Triangular shapes are used in construction to create rigid
structures.
Quadrilaterals, on the other hand, are not rigid structures. In the diagram
below, you can see how a rectangle can be pushed to form a parallelogram. All
sides remain the same length, but the interior angles change.
push
A rectangle, therefore, is not a rigid structure. Similarly, any polygon with
four or more sides is not a rigid structure. As stated above, however, a triangle
is a rigid structure. If you push a triangle, the shape c_annot change without
changing the side lengths.
push
Due to the rigid structure of triangles, they are a common shape in
construction of roof supports and tall towers.
In the previous lesson, you studied different types of triangles, which
included calculating the side lengths and angle measurements of right-angled
triangles. In this lesson, you will learn how to find the lengths of sides and
measurements of angles of triangles that are not right-angled triangles.
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Trigonometry —Sine Law and Cosine Law
Non-right triangles are triangles that do not have a 90° angle. The
trigonometric ratios of sin, cos, and tan are used to find the unknown
measurements in right triangles only To solve for the unknown
measurements in triangles that do not have a right angle, you need different
formulas. One of the foil tulas you can use is the Sine Law, also known as the
Law of Sines.
It may be helpful for you to add this formula to your resource sheet. You
should also include the situations in which the Sine Law can be used.
The Sine Law in words is: "The ratio of the sine of any angle over its opposite
side is a constant."
The Sine Law is written as follows:
sin ZA sin ZB sin ZC
a
Or
a
sin LA ▪ sin LB - sin ZC
As long as you know both an angle and its opposite side, the Sine Law can
help you.
The Sine Law can be used in two situations.
Situation One
Given two angles of a triangle and one of its sides, you can solve a non-right
triangle by using these steps:
1. Find the third angle by subtracting the sum of the two given angles from
1800.
2. Find the two missing sides using the Sine Law.
You may remember that solving a triangle means finding the lengths of ALL
the missing sides, and the measurements of ALL the missing angles.
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Trigonometry —Sine Law and Cosine Law
Example 1
Find the lengths of side a.
A
t't
Example 2
Solve the triangle by finding the lengths of sides a and b, and the
measurement of LB.
A
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Trigonometry—Sine Law and Cosine Law
Situation Two
If you are given the lengths of two sides and the measurement of one angle
of a triangle, you can use the Sine Law, but only if the angle you are given is
opposite one of the given sides.
Example 3
Explain in which of the following triangles you could use the Sine Law.
a)
Example 4
Find the measurements of LA and LC of AABC, given the diagram below.
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Trigonometry —Sine Law and Cosine Law
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Example 5
Suppose you are the pilot of a commercial airliner. You find it necessary to
detour around a group of thundershowers. You turn at an angle of 21° to
your original path, fly for a while, turn, and intercept your original path at an
angle of 35°, 70 kilometres from where you left it.
How much further did you have to go because of the detour?
947,11' 3"
r
N. ,7
70
Example 6
Two observers on boats have located a sunken ship using sonar equipment.
The sunken ship is in line between the two observers. The angle of depression
from Observer A to the sunken ship is 40°, and the angle of depression from
Observer B to the sunken ship is 36°. The distance between the observers is
450 metres. How far is the ship below the surface of the water? Round your
answer to the nearest metre.
B (water surface)
‘N;P
/01
S
(sunken ship)
ve
Note that the depth is the perpendicular distance from the sunken ship to the
surface of the water.
1. LA, LB, and LC are all angles in an isosceles triangle. If LA = 40° and LB
and LC are base angles in the triangle, what is the measure of LC?
2. Which polygon has all equal sides and all interior angles equal to 90°?
3. LA and LB are opposite interior angles in a parallelogram. If LA = 38°,
what is the measure of LB?
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Trigonometry —Sine Law and Cosine Law
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1. Find the length of side b.
a)
b)
2. Find the measurement of angle C.
a)
3. Find the length of AR
A
*
S if\
c
‘-t
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•
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Trigonometry—Sine Law and Cosine Law
4. All and Beth are both looking at a hot air balloon. For Alf the angle of
elevation is 40°, and for Beth the angle of elevation is 58°. How high is the
balloon above ground if the distance between All and Beth is 600 metres?
In the diagram, A represents All, B represents Beth, H represents the hot air
balloon, and HC is the distance of the balloon above the ground.
k-21 Yu.
600 in
5. Fire ranger Jarnal is 20 km due west of fire ranger Raj. They spot a fire and
report the sightings as follows:
m Jamal reports that the fire is N 470 E of his station.
m Raj reports that the fire is N 20° W of his station.
How far is the fire from Jamal's station?
>8
Write the values of LAJR and LARJ into the diagram before you solve the
problem.
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Trigonometry Sine Law and Cosine Law
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The Cosine Law can be used to solve certain non-right triangles. just as the
Sine Law could only be used in certain situations, it is also true that the
Cosine Law can only be used in certain situations. Make sure you add this
formula to your resource sheet, along with the situations in which you can
use the Cosine Law.
The Cosine Law is written as follows for any triangle, ABC:
a2 = b2 + C2 - 2bc cos LA
b2 (12 c2 _ 7a c cos GB
c2 = a2 + b2 - 2ab cos GC
Note that there axe three versions of this formula, but there is a pattern.
Notice the side squared on the left of the formula always corresponds to the
angle on the right side of the formula.
As with the Sine Law, in order for the Cosine Law to work, you must label
your triangles correctly, where a is opposite GA, b is opposite GB, and so on
Just like the Sine Law, the Cosine Law can be used in only two situations.
Situation One
If you are given two sides and the angle between them you can solve the
triangle using the following steps.
1. Calculate the length of the missing side using the Cosine Law.
2. If required, you can use the Sine Law to calculate the measure of the
smallest of the unknown angles. The Sine Law may be used more accurately
with smaller angles.
3. If required, you can find the third angle using the fact that the sum of angles
in a triangle is 180'.
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Trigonometry—Sine Law and Cosine Law
Example 1
Given the triangle measurements as shown, find the measure of side a.
Example 2
Find the length of side :v.
Situation Two
If you are given three sides of a non-right triangle, you can solve the triangle
using the following steps:
1. Determine the measure of one of the larger angles using the Cosine Law
(larger angles are opposite longer sides).
2, Calculate the measure of the smallest angle using the Sine Law.
3. Find the third angle using the fact that the sum of angles in a triangle is
180°.
ct l‘
L, _
kAl1/14 (
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H
t
Trigonometry -Sine Law and Cosine Law
it
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:26
USC
Example 1
Find the measure of the smallest angle in triangle ABC, if a = 7, b = 8, and
c =5.
C.
7
2-
/1-
,
* Example 2
Solve for angles LA, LB, and LC in the following diagram.
'2-
2-
t-, ( a
66.7
51
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Trigonometry—Sine Law and Cosine Law
Which formula should you use to solve the following problems—the Sine
Law or the Cosine Law? Do not solve the problems.
a) Find the length of zu.
LI) Find the length of t.
c) Find the measure of LA.
d) Find the measure of /D.
15
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Trigonometry—Sine Law and Cosine Law
2. Find the measure of the side or angle indicated.
a) Find side b.
b) Find angle A.
A
c) Find side a.
„Jot.t AB
(,(
1
101.1
C--
128.7
3. The arms of a "jaws" machine used for prying open crushed vehicles in
accidents are 75 cm long. The jaws are joined at one end and open at various
angles. If the angle formed by the arms is 55.6°, how far apart are the tips?
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Trigonometry—Sine Law and Cosine Law
The posts of a hockey goal are 2.0 m apart. A player attempts to score by
shooting the puck along the ice from a point 6.5 m from one post and 8.0
m from the other post. Within what angle, 9, must the shot be made? (Give
your answer to the nearest degree.)
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