Applications to Simple Problems in Two or Three Dimensions

Understanding Plane Geometry and Trigonometry

Plane geometry involves the study of shapes in two dimensions (2D) such as circles, triangles, and polygons.
2D problems involving trigonometry often require the use of sine, cosine, and tangent ratios to solve for unknown sides and angles in right triangles.
In 2D geometry, it’s also common to use the Pythagorean theorem (a² + b² = c²) when dealing with right-angled triangles.
Remember to label all sides and angles in your diagrams; this makes it easier to identify which trigonometric functions to use.

Solving Trigonometry Problems in Three Dimensions (3D)

When moving to 3D geometry, we add an extra dimension on the z-axis.
Problems in 3D often involve shapes like cubes, spheres, and tetrahedrons, where you are asked to find lengths of sides, angles, volumes, or surface areas.
These problems may require the use of spherical or cylindrical coordinates, and you’ll often use both the Pythagorean theorem and trigonometric relationships in your solutions.
When visualising these problems, it can be helpful to draw a 3D diagram.

Bearings and Trigonometry

Bearings are often used to describe angles in navigational contexts.
A bearing is an angle measured clockwise from north and is usually given as a three-digit number.
Trigonometry often comes into play when finding the bearing between two points given certain information about their relationship, such as distance and angles from a certain point.

Applications in Physics and Engineering

Remember that trigonometry is widely used in physics and engineering, such as for making calculations related to waves, architecture, and structural designs.
One example is to calculate the resultant force of several forces acting at different angles.
Another application could be to calculate the height of an object using its shadow and the angle of the sun.

Remember to Practise

Work through a variety of practice problems. This will help you identify the type of problem, select the right tools for solving it, and gain confidence in executing each step.
Finally, challenge yourself with complex problems. They’re an opportunity to apply multiple skills and reinforce your understanding of trigonometry principles. Always make sure to check your work for potential mistakes and inaccuracies.