Law of sines
In
trigonometry, the law of sines is a statement about arbitrary
triangles in the plane. If the sides of the triangle are
a,
b and
c and the
angles opposite those sides are
A,
B and
C, then the law of sines states:
where
R is the radius of the triangle's
circumcircle. This formula is useful to compute the remaining sides of a triangle if two angles and a side are known, a common problem in the technique of
triangulation. It can also be used when two sides and one of the non-enclosed angles are known; in this case, the formula may give two possible values for the enclosed angle.
Encyclopedia
In
trigonometry, the
law of sines is a statement about arbitrary
triangles in the plane. If the sides of the triangle are
a,
b and
c and the
angles opposite those sides are
A,
B and
C, then the law of sines states:
where
R is the radius of the triangle's
circumcircle. This formula is useful to compute the remaining sides of a triangle if two angles and a side are known, a common problem in the technique of
triangulation. It can also be used when two sides and one of the non-enclosed angles are known; in this case, the formula may give two possible values for the enclosed angle. When this happens, often only one result will cause all angles to be less than 180°; in other cases, there are two valid solutions to the triangle .
It can be shown that:
where
s is the
semi-perimeter,The ambiguous case
When using the law of sines to solve triangles, under special conditions there exists an ambiguous case where two separate triangles can be constructed .
Given a general triangle
ABC, the following conditions would need to be fulfilled for the case to be ambiguous:
- The only information known about the triangle is the angle A and the sides a and b, where the angle A is not the included angle of the two sides .
- The angle A is acute .
- The side a is shorter than the side b .
- The angle B is not a right angle .
Given all of the above premises are true, the angle
B may be acute or obtuse; meaning, one of the following is true:
-
OR
-
Derivation
Make a triangle with the sides
a,
b, and
c, and angles
A,
B, and
C. Draw a line from the angle
C to the side across
c so that it divides the original triangle into two right angle triangles. Mark the length of this line
h.
It can be observed that:
and
Therefore:
and
Doing the same thing with the line drawn between angle
A and side
a will yield:
Full proof:Make a triangle
ABC with sides
a,
b,
c and the
? angle at
C.
Make an axis through the center of
b and another through the
c side. Mark the point of intersection of the axis
S.
Draw a circle
k with its center in
S with the radius
r = |SA| = |SB| = |SC| .
Through the medial angle law, the angle at S is 2*
?.
Thus, it can be observed that:
or:
and then
Applying cyclic permutation:
Examples
Here is an example of how to solve a problem using the law of sines:
Given: side
a = 10, side
c = 7, and angle
C = 30 degrees
Using the law of sines, we know that
Plugging in the given values, we find that
Simplifying, the sine of angle
A is equal to 5/7, or approximately 0.714. Thus, angle
A is equal to 45.58 degrees.
Or another example of how to solve a problem using the law of sines:
If two sides of the triangle are equal to
R and the length of the third side, the chord, is given as 100' and the angle
C opposite to the chord is given in
degrees, then angle
A = angle
B = and
or or This is North American
railroad surveying practice.
See also
External links
