Trigonometry – Right-Angled Triangles - Finding the length of a side

Trigonometry – Right-Angled Triangles - Finding the Length of a Side

Fundamental Concepts

• Trigonometry is a branch of mathematics that deals with the relationships between side lengths and angles of triangles. The right-angled triangles are the core highlight in trigonometry.

• In a right-angled triangle, we have: 1) the Hypotenuse (the longest side opposite the right angle), 2) the Adjacent (the side along with the angle we are interested in), and 3) the Opposite (the side opposite to the angle we are interested in).

Trigonometric Ratios

• The three main trigonometric ratios which relate the sides of a right-angled triangle to its angles are: Sine (Sin), Cosine (Cos), and Tangent (Tan).

• Their definitions in a right-angled triangle are as follows: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

• The acronym SOHCAHTOA is a helpful mnemonic to remember these relationships.

Finding the Length of a Side

• If the length of one side and an angle (other than the right angle) of a right-angled triangle is known, trigonometry can be used to find the length of another side.

• Use the correct trigonometric ratio according to which sides are known and which side’s length is to be found.

• After substituting the known values in the correct ratio, rearrange the equation to solve for the unknown length.

Useful Tips

• Always ensure the presented triangle is a right-angled triangle before attempting to use trigonometry.

• Be comfortable with converting between degrees and radians, as angles can be given in either unit in the problems.

• Practice is key when it comes to mastering the application of trigonometric ratios.

This robust understanding of right-angled triangles, trigonometric ratios, and their applications in finding side lengths will be paramount in tackling geometry problems. This knowledge will also be critical for more complex geometric and trigonometric problems.