504 CHAPTER 7 Analytic Trigonometry For example, the following are identities: x x x x x x x 1 2 1 sin cos 1 csc 1 sin 2 2 2 2 ( ) + = + + + = = The following are conditional equations: x x x x 2 5 0 sin 0 sin cos + = = = True only if =− x 5 2 True only if π =x k k , an integer True only if π π = + x k 4 2 or π π = + x k k 5 4 2 , an integer Below are the trigonometric identities that have been established so far. Quotient Identities tan sin cos cot cos sin θ θ θ θ θ θ = = Reciprocal Identities csc 1 sin sec 1 cos cot 1 tan θ θ θ θ θ θ = = = Pythagorean Identities sin cos 1 tan 1 sec cot 1 csc 2 2 2 2 2 2 θ θ θ θ θ θ + = + = + = Even–Odd Identities sin sin cos cos tan tan csc csc sec sec cot cot θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) − = − − = − = − − = − − = − = − These are called the basic trigonometric identities . These identities should not merely be memorized, but should be known (just as you know your name rather than have it memorized). In fact, minor variations of a basic identity are often used. For example, sin 1 cos or cos 1 sin 2 2 2 2 θ θ θ θ = − = − might be used instead of sin cos 1. 2 2 θ θ + = For this reason, among others, it is very important to know these relationships and be comfortable with variations of them. 1 Use Algebra to Simplify Trigonometric Expressions The ability to use algebra to manipulate trigonometric expressions is a key skill that one must have to establish identities. Four basic algebraic techniques are used to establish identities: • Rewriting a trigonometric expression in terms of sine and cosine only • Multiplying the numerator and denominator of a ratio by a “well-chosen 1” • Writing sums of trigonometric ratios as a single ratio • Factoring
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