Trigonometry studies the relation between angles in a triangle and the ratios of its sides. In MAT 1175, we restrict our study to trigonometry of right triangles. Trigonometry of non-right or oblique triangles will be done in MAT 1275.
In a right triangle, the 2 sides opposite the acute angles are called the legs. The side opposite the right angle is the hypotenuse. We take the point of view of one of the acute angles (marked x): the opposite side (O) is the leg opposite to the angle and the adjacent side (A) is the leg next to the angle.
The 3 main trigonometric functions — sine, cosine and tangent — are defined as follows:
A mnemonic that may help you remember is SOH-CAH-TOA.
- SOH: Sine of angle is Opposite over Hypotenuse;
- CAH: Cosine of angle is Adjacent over Hypotenuse;
- TOA: Tangent of angle is Opposite over Adjacent.
Trigonometry works because triangles with the same set of angles are similar and hence, the ratios between the sides will always be the same. For example, the ratio of the opposite leg to the hypotenuse will be the sin A for any right triangle with acute angle A.
Here is the introductory Khan Academy video:
Unfortunately, the other KA videos use radians which won’t be introduced until MAT 1275. [Please make sure your calculators are in degree mode!]
To solve a right triangle means finding missing sides and angles of a triangle using trigonometry and possibly the Pythagorean theorem, which we will see later in the semester but which for many of you may already be familiar. Given one side and an angle, regular trig functions can be used to solve. In contrast, inverse trig functions are needed to solve if 2 sides are given. Below you will find 2 parts to a more difficult video. The first part reviews how to solve a triangle then has an example using the Pythagorean theorem:
The second part focuses on applications. Pay close attention to the definitions for angles of elevation and depression and the terminology and conventions related to navigation. The last example, involving navigation, uses the Pythagorean theorem.
A historical note: trigonometry, while compiled in ancient Greece by Hipparchus, was developed extensively by the Hindu and Muslim civilizations and also by the Chinese. (“sine” can be traced back to the Sanskrit word jya which means “bowstring”.) Beginning with the Renaissance and continuing into the Age of Enlightenment (17th century), trigonometry received further development, culminating with the 18th century Swiss mathematician and physicist Leonhard Euler (rated one of the top 10 mathematicians of all time).