Fundamental Identities and Example Problems in Trigonometry

Section 7.1: Fundamental Identities discusses several important relationships in trigonometry based on various principles.

Reciprocal identities illustrate the fundamental relationships for trigonometric functions, where the cosecant is defined as the reciprocal of sine, the secant as the reciprocal of cosine, and the cotangent as the reciprocal of tangent. Specifically, these can be expressed as follows: cosec = (\frac{1}{\text{sin}}), sec = (\frac{1}{\text{cos}}), and cot = (\frac{1}{\text{tan}}).

Next, the quotient identities express tangent and cotangent in terms of sine and cosine. The tangent of an angle (\theta) is defined as (\tan(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}), while the cotangent is given by (\text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)}).

The Pythagorean identities are key relationships derived from the Pythagorean theorem. They include the fundamental identity (\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1), along with two additional identities: (\tan^2(\theta) + 1 = \text{sec}^2(\theta)) and (1 + \text{cot}^2(\theta) = \text{csc}^2(\theta)).

There are also even/odd identities that describe the symmetry properties of trigonometric functions. The sine and cosecant functions are odd, which means (\text{sin}(-\theta) = -\text{sin}(\theta)) and (\text{csc}(-\theta) = -\text{csc}(\theta)). On the other hand, the cosine and secant functions are even, satisfying (\text{cos}(-\theta) = \text{cos}(\theta)) and (\text{sec}(-\theta) = \text{sec}(\theta)). The tangent and cotangent functions are odd as well, fulfilling (\tan(-\theta) = -\tan(\theta)) and (\text{cot}(-\theta) = -\text{cot}(\theta)).

Example problems are provided to illustrate the application of these identities. For instance, given that (\text{cos}(\theta) = \frac{5}{6}) in quadrant I, the Pythagorean identity can be used to find that (\text{sin}(\theta) = \frac{\sqrt{11}}{6}). In another example, starting with (\text{tan}(\theta) = -\frac{\sqrt{7}}{2}) and (\text{sec}(\theta) > 0), one can derive that (\text{sec}(\theta) = \frac{\sqrt{11}}{2}), leading to the calculation of (\text{cos}(\theta) = \frac{2}{\sqrt{11}}).

Further examples illustrate how to find sine from cosine, applying the identity (\text{cos}(-\theta) = \text{cos}(\theta)) before using the Pythagorean theorem to compute (\text{sin}(\theta)). Lastly, an example about finding all trigonometric functions given (\text{cos}(\theta) = 1) in quadrant I shows the systematic approach of deriving results through the fundamental identities.