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By lettting c1 and c2 be the legs of a right triangle, with angle φ adjacent to c2, we get and Substituting into (2), we get Using a trigonometric identity, we get: (3) Let and . Substituting this into (3) gives Analytic Trigonometry Using Fundamental Identities Some of the fundamental trigometric identities are those derived from the Pythagorean Theorem. These are defined using a right triangle: By the Pythagorean Theorem, A2 + B2 = C2 {1} Dividing through by C2 gives {2} We have already defined the sine of a in this case as A/C and the cosine of a as B/C.

It is important to remember that merely verifying an identity or altering an expression is not an end in itself, but rather that identities are used to simplify expressions according to the task at hand. Trigonometric expressions can always be reduced to sines and cosines, which are more managable than their inverse counterparts. To verify an identity: 1) Always try to reduce the larger side first. 2) Sometimes getting all trigonometric functions on one side can help. 3) Remember to use and manipulate already existing identities.

3) Remember to use and manipulate already existing identities. The Pythagorean identities are usually the most useful in simplifying. 4) Remember to factor if needed. 5) Whenever you have a squared trigonometric function such as Sin^2(t), always go to your pythagorean identities, which deal with squared functions. Easy example 1 / cot(t) = sin(t) / cos(t) 1/cot(t)= tan(t) ,so tan(t) = sin(t) / cos(t) tan(t) is the same as sint(t)/cos(t) and therefore can be rewritten as sin(t) / cos(t) = sin(t) / cos(t) Identity verified Solving-Trigonometric Equations Trigonometric equations involve finding an unknown which is an argument to a trigonometric function.