Application of Obliques in Geometry

Application of Obliques

Law of Sines

Note that the law of sines says that three ratios are equal.

If you know two angles and the side opposite one of them, then you can determine the side opposite the other one of them.

If you know two sides and the angle opposite one of them, then you can almost determine the angle opposite the other one of them.

The Ambiguous Case

An interesting problem arises when two sides and an angle opposite one of them are known.

The possible solutions depend on whether the given angle is acute or obtuse. When the angle is acute, five possible solutions exist. When the angle is obtuse, three possible solutions exist.

Obtuse

Acute

1. If the side opposite the given angle is shorter than the other given side and less than a certain length, then no solution exists.

If the side opposite the given angle is less than the other given side, then there is no solution, and no triangle is determined.

2. If the side opposite the given angle is shorter than the other given side, exactly one solution exists, and a right triangle is determined.

If the side opposite the given angle is equal to the other given side, then there is no solution, and, again, no triangle is determined.

3. If the side opposite the given angle is shorter than the other given side, but longer than in case (2), then two triangles are determined, one in which A = x degrees, and one in which A = 180 o - x degrees.

Law of Cosines

4. If the side opposite the given angle is equal in length to the other given side, then A = B , and one isosceles triangle is determined.

If the side opposite the given angle is greater than the other given side, then exactly one triangle is determined.

5. If the side opposite the given angle is longer than the other given side, then one triangle is determined.

The law of cosines relates the three sides of the triangle to one of the angles.

Knowing the oblique angle and its two adjacent sides, you can determine the opposite side using the law of cosines.

For example:

C = 60°

a = 5

b = 8;

therefore, c = 7.

Oblique Angles

Examples:

In Solving

Oblique Triangles

Two theorems of geometry give useful laws of trigonometry that will aid in solving for oblique triangles.

These are called the “law of cosines” and the “law of sines.” There are other “laws” that used to be used, but since the common use of calculators, these two laws are enough.

AB is a line 652 feet long on one bank of a stream, and C is a point on the opposite bank. A = 53° 18', and B = 48° 36'. Find the width of the stream from C to AB.

In a triangle ABC, a = 700 feet, B = 73° 48', and C = 37° 21'. If M is the middle point of BC find the length of AM, and the angles BAM

and MAC.

As stated before, angles that are not right angles or a multiple of a right angle are called oblique angles.