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Law of Sine and Law of Cosine

Law of Sine

The figure below show an acute triangle ABC, its three vertex A, B and C opposite three sides a, b and c respectively. What is the relation among the sides and the angles in the triangle? (Note: a capital letter represents the vertex and a small letter represents the length of a side in the triangle. )

Show an acute triangle used to describe the law of Sine.

Now we draw a line that pass through the pint C and perpendicular to AB at the point D. (D is the perpendicular point.)

How to prove the law of Sine?
In right triangle ACD, use the definition of Sine, sin A = CD/AC = CD/b
so CD = b sin A
In right triangle BCD, sin B = CD/BC = CD/a
so CD = a sin B
Then b sin A = a sin B
(Both sides of the equation divide by sin B sin A)
so b/sin B = a/sin A
The formula above tell us, in triangle ABC, the ratio of b to sin B equals the ratio of a to sin A (Note: in triangle ABC, side b opposite the angle B and the side a opposite the angle A).

Now we will find the relation between the side c and the angle C.

Prove the law of sine.
angle C = angle 1 + angle 2
sin C = sin(angle1 + angle2) = sin(angle1) cos(angle2) + cos(angle1) sin(angle2)
sin(angle1) = BD/BC = BD/a
cos(angle2) = CD/AC = CD/b
cos(angle1) = CD/BC = CD/a
sin(angle2) = AD/AC = AD/b
Then sin C = BD/a × CD/b + CD/a × AD/b = [CD/ab] × (BD + AD)
Note: BD + AD = AB = c
Then sin C = [CD/ab] × c
Note: sin B = CD/a
So sin C = [sin B/b] × c
Both sides divide by the side c
So sin C/c = sin B/b
So c/sin C = b/sin B
Therefore, we get the law of Sine:
a/sin A = b/sin B = c/sin C

The law of Sine describe the relation between three sides and the Sine of their corresponding angles. If we know two angles and a side, we can find other angle in the triangle from the sum of a triangle theorem. Then apply the law of Sine to calculate the other two sides. If we know any two sides and the angle opposite one of the sides, apply the law of Sine, we can find the value of Sine of the angle of other side opposite, from here to find other sides and angles.

What is the relation among the sides and angles in an obtuse triangle?

The figure below show an obtuse triang (90o < angle C < 180o). Now we prove the law of Sine.

Prove the law of sine in an obtuse triangle.

Draw a line passing through the point C and perpendicular to AB at point D (D is the perpendicular point.)

Prove the law of sine in an obtuse triangle.
In the figure above, in the right triangle BCD, sin B = CD/BC = CD/a,
so CD = a sin B
In the right triangle ACD, sin A = CD/AC = CD/b,
so CD = b sin A
So a sin B = b sin A,
Both sides of the equation divide by sin A sin B
so a/sin A = b/sin B

Now we will find the relation between sin C and c

Extended the BC to the left and draw a line passing through the point A and perpendicular to the extended BC line at the point E (E is the perpendicular point.)

Prove the law of sine in an obtuse triangle.
In the figure above, since BE is a straight line, so the angle BCE = 180o
angle C = 180o - angle 1
sin C = sin(180o - angle1) = sin180o cos(angle1) - cos180o sin(angle1)
Since sin180o = 0, cos180o = -1
So sin C = sin(angle1)
In right triangle ACE, sin(angle1) = AE/AC = AE/b,
so AE = b sin(angle1) = b sin C
In right triangle ABE, sin B = AE/AB = AE/c,
so AE = c sin B
So b sin C = c sin B
Both sides of the equation divide by sin B sin C
So b/sin B = c/sin C
Therefore, we get,
a/sin A = b/sin B = c/sin C

Thus we prove that the law of Sine satisfy the obtuse triangle.

The Law of Sine

Show the Law of Sine.

Apply the law of Sine to solve problems

Example 1

Question:
In triangle ABC, a = 49, B = 75o and C = 45o, what is the value of c?
Solution:
A = 180o - (B + C) = 180o - (75o + 45o) = 180o - 120o = 60o
The degree measure of the angle A is 60o
Apply the law of Sine
Law of sine eample 1 solution.
The value of c is 40.
the graph of law of sine example 1.

Example 2

Question:
In triangle ABC, c = 120, C = 105o and b = 88, find a, A and B.
Solution:
sin C = sin105o
= sin(180o - 75o)
= sin75o
= sin(45o + 30o)
= sin45o cos30o + cos45o sin30o
Law of Sine example2 fig2
= 0.9659
Apply the law of Sine
Find the value of sin B using the law of Sine example2 solution.
Because sin B = 0.7083, so B = 45o
A = 180o - (B + C)
= 180o - (45o + 105o)
= 180o - 150o
= 300
Apply the law of Sine
Find the length of the side opposite the angle A using the law of Sine example2 solution.
Thus, we get the triangle,
Apply the law of sine to get the triangle.
In triangle ABC, because C > B > A, so c > b > a

Law of Cosine

If we know two sides and the angle between them, how to find the third side in the triangle? For example, if we know the sides b, c and the angle A, how to find the length of the side a?

triangle to describe the law of cosine.

In xy-plane, let the point A be the origin and the side AB lies on the a-axis. In the figure below, the coordinate of the point A is (0, 0), the coordinate of the point B is (x1, 0), [note: the y-coordinate of the point B is 0, because the point B lies on the x-axis], the coordinate of the point C is (x2, y2). Now we find the length of the side a.

prove the law of Cosine
use the xy-plane to prove the law of cosine.
use the xy-plane to prove the law of cosine figure 2.
Note: cos A = x2/square root of (x22 + y22)
Thus, we get the law of Cosine, a2 = b2 + c2 - 2bc cosA

The law of Cosine

triangle to describe the law of cosine.
The figure show the law of the Cosine.
If we know three sides in a triangle, how to find the angles in this triangle? From the law of Cosine, we get the following formula.
The inference law getting from the law of Cosine.

Example 3

Question:
In triangle ABC, the length of the side a = 36, B = 45o and the length of the side c = 40, find the triangle.
Solution:
Find the length of the side b
b2 = a2 + c2 - 2ac cos B
= 362 + 402 - 2 × 36 × 40 cos45o
= 1296 + 1600 - 2880 × (square root of 2)/2
= 860
b = square root of 860 = 29.32
Find the angle B
Example of using the law of Cosine.
Find the angle C
Example of using the law of Cosine.
The triangle shown below
The triangle that use the law of cosine