Oblique Triangles and Coordinate Geometry: Solving Triangles and Finding Coordinates, Exercises of Geometry

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Math For Surveyors

James A. Coan Sr. PLS

Topics Covered

1. The Right Triangle

2. Oblique Triangles

3. Azimuths, Angles, & Bearings

4. Coordinate geometry (COGO)

5. Law of Sines

6. Bearing, Bearing Intersections

7. Bearing, Distance Intersections

The Right

Triangle

Side Adjacent (b)

Side Opposite (a) A

B

C

CosA b c

= TanA a b

SineA = a c

=

CscA

c a

= SecA^

c

b

= (^) CotA

b a

=

SinA • c = a

a

SinA

= c

The Right Triangle

Example:

CosA • c = b

b CosA

= c

TanA • b = a a TanA

= b

SinA

a c

CosA b c

=

TanA

a b

Oblique Triangles

An oblique triangle is one that does

not contain a right angle

a

Sin A

b

Sin B

c

Sin C

The Law of Sines

Oblique Triangles

A B

C

b^ a

c

Oblique Triangles

The law of Cosines

a^2 = b^2 + c 2 - 2bc Cos A

A B

C

a

b

c

Oblique Triangles

When angle A is obtuse (more than 90°) and side a is shorter than or equal to side c, there is no solution.

A B

C

b a

c

Oblique Triangles

When angle A is obtuse and side a is greater than side c then side a can only intersect side b in one place and there is only one solution.

A

B

C

a b

c

Oblique Triangles

When angle A is acute, and the height is given by the formula h = c Cos A, and side a is less than h, but side c is greater than h, there is no solution.

A B

b

c

a

h

Oblique Triangles

When angle A is acute and side a = h, and h is less than side c there can be only one solution

A B

C

a = h

b

c

Azimuth

Angles

Bearings

Azimuth, Angles, & Bearings

Azimuth:

An Azimuth is measured clockwise from North.

The Azimuth ranges from 0° to 360°