Oblique Triangles

An oblique triangle does not have a right angle and can also be classified as an acute triangle or an obtuse triangle.

To solve oblique triangles, use the laws of sine and cosine. There are four different potential scenarios:

1. Solve a Triangle Knowing: One Side and Two Adjacent Angles.


Solving Oblique Triangles

Solving Oblique Triangles

Solving Oblique Triangles

Oblique Triangle

Example

Solve the oblique triangle with the following data: a = 6 m, B = 45° and C = 105°.

Oblique Triangle

Trigonometry Exercise

Trigonometric Operations

Trigonometric Solution


2. Solve a Triangle Knowing Two Sides and an Included Angle.

Solving Oblique Triangles

Solving Oblique Triangles

Solving Oblique Triangles

Triangle

Example

Solve the oblique triangle with the following data: a = 10 m, b = 7 m and C = 30°.

Triangle

Trigonometry Exercise

Trigonometric Operations

Trigonometric Operations

Trigonometric Solution


3. Solve a Triangle Knowing Two Sides and the Opposite Angle.

Solving Oblique Triangles

sin B > 1. No solution

sin B = 1. One solution

sin B < 1. One or two solutions

Triangle

1. sin B > 1. No solution

Solve the triangle with the following data: A = 30°, a = 3 m and b = 8 m.

Solving Oblique Triangles

Since the sine of an angle can never be greater than 1, the problem has no solution. The drawing above shows the impossibility of the situation.


2. sin B = 1. One Solution: Right Triangle

Solve the triangle with the following data: A = 30°, a = 3 m and b = 6 m.

Solving Oblique Triangles




Solving Oblique Triangles

Triangle Solution

Solving Oblique Triangles

Trigonometric Solution


3. sin B < 1. One or Two Solutions

Solve the triangle with the following data: A = 60°, a = 8 m and b = 4 m.

Solving Oblique Triangles

Trigonometric Operations

Trigonometric Operations

Trigonometric Solution


Solve the triangle with the following data: A = 30°, a = 3 m and b = 4 m.

Trigonometry Exercise

Trigonometric Operations

Trigonometric Operations

Trigonometric Operations

Trigonometric Operations

Trigonometric Solution


4. Solve a Triangle Knowing Two Sides and the Opposite Angle.

Solving Oblique Triangles

Solving Oblique Triangles

Solving Oblique Triangles

Triangle

Solve the triangle with the following data: a = 15 m, b = 22 m and c = 17 m.

Trigonometry Exercise

Trigonometric Operations

Trigonometric Solution