Trigonometry

One immediate application for trigonometric functions is the simple determination of the dimensions of a right triangle, also known as the solution of a triangle, when only a few are known. For example, if the sides of a right triangle are known, then the magnitude of both acute angles can be found. Suppose we have a right triangle whose sides are 2 in (5 cm) and 4.7 in (12 cm), and whose hypotenuse is 5.1 in (13 cm). The unknown angles could be found by using any trigonometric function. Since the sine of one of the angles is equal to the length of the opposite side divided by the hypotenuse, this angle can be determined. The sine of one angle is 5/13, or 0.385. With the help of a trigonometric function table or calculator, it will be found that the angle which has a sine of 0.385 is 22.6°. Using the fact that the sum of the angles in a triangle is 180°, we can establish that the other angle is 180° - 90° - 22.6° = 67.4°.

In addition to solving a right triangle, trigonometric functions can also be used in the determination of the area when given only limited information. The standard method of finding the area of a triangle is by using the formula, area = 1/2b (base) × h (altitude). Often, the altitude of a triangle is not known, but the sides and an angle are known. Using the side-angle-side (SAS) theorem, the formula for the area of a triangle then becomes, area = 1/2 (one side) × (another side) × (sine of the included angle). For a triangle with sides of 5 cm and 3 cm respectively and an included angle of 60°, the area of the triangle would be equal to 1/2 × 5 × 3 × sin 60° = 13 cm².

The formula for the area of a triangle leads to an important concept in trigonometry known as the Law of Sines which says that for any triangle, the sine of each angle is proportional to the opposite its opposite side, symbolically written in triangle ABC as,

Using the Law of Sines, we can solve any triangle if we know the length of one side and magnitude of two angles, or two sides and one angle. Suppose we have a triangle with angles of 45° and 70°, and an included side of 15.7 in (40 cm). The third angle is found to be 180° 45° - 70° = 65°. The unknown sides, x and y, are found with the Law of Sines because

The lengths of the unknown sides are then x = 12.29 in (31.2 cm) and y = 16.35 in (41.5 cm).

The Law of Sines can not be used to solve a triangle unless at least one angle is known. However, a triangle can be solved if only the sides are known by using the Law of Cosines which is stated in triangle ABC, c² = a² + b² - 2ab cos C, or can be written

which is more convenient when using only the sides to solve a triangle. As an example, consider a triangle with sides equal to 2 in, 3.5 in, and 3.9 in (5 cm, 9 cm, and 10 cm). The cosine of one angle would be equal to (5² + 9² - 10²)/(2 × 59) = 0.067, which corresponds to the angle 86.2°. Similarly, the other two angles are found to be 29.9°and 63.9°.