Trigonometry

In addition to the reciprocal relationships of certain trigonometric functions, two other types of relationships exist. These relationships, known as trigonometric identities, include cofunctional relationships and Pythagorean relationships. Cofunctional relationships relate functions by their complementary angles. Pythagorean relationships relate functions by application of the Pythagorean theorem.

The sine and cosine of an angle are considered cofunctions, as are the secant and cosecant, and the tangent and cotangent.

The Pythagorean theorem states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. For a triangle with sides of x and y and a hypotenuse of z, the equation for the Pythagorean Theorem is x²+ y² = z². Applying this theorem to the trigonometric functions of an angle, we find that sin² θ + cos² θ = 1. Similarly, 1 + tan² θ = sec² θ and 1 + cot² θ = csc² θ . The terms such as sin² θ or tan² θ traditionally have meant (sin θ) × (sin θ) or (tan θ) × (tan θ).

In some instances, it is desirable to know the trigonometric function of the sum or difference of two angles. If we have two unknown angles, θ and &NA;, then sin ( θ + &NA;) is equal to sin θcos &NA; + cos θsin &NA;. In a similar manner, their difference, sin( θ- &NA;) is sin θcos &NA; cos θsin &NA;. Equations for determining the sum or differences of the cosine and tangent also exist and can be stated as follows:

cos( θ ± &NA; ) = cos θcos &NA; ± sin θsin &NA; tan ( θ ± &NA; ) = (tan θ ± tan &NA;)/(1 ± tan θtan &NA;)

These relationships can be used to develop formulas for double angles and half angles. Therefore, the sin 2 θ = 2sin θcos θ and cos 2 θ = 2cos² θ - 1 which could also be written cos θ2 = 1 - 2sin² θ.