SECTION 6.3 Properties of the Trigonometric Functions 419 The identities in (5), (6), and (7) are referred to as the Pythagorean Identities . Collectively, the identities in (2), (3), and (5)—(7) are referred to as the Fundamental Identities . Fundamental Identities • θ θ θ = tan sin cos • θ θ θ = cot cos sin • θ θ = csc 1 sin • θ θ = sec 1 cos • θ θ = cot 1 tan • θ θ + = sin cos 1 2 2 • θ θ + = tan 1 sec 2 2 • θ θ + = cot 1 csc 2 2 Finding the Exact Value of a Trigonometric Expression Using Identities Find the exact value of each expression. Do not use a calculator. (a) ° − ° ° tan20 sin20 cos20 (b) π π + sin 12 1 sec 12 2 2 Solution EXAMPLE 4 (a) ° − ° ° = ° − ° = tan20 sin20 cos20 tan20 tan20 0 ↑ θ θ θ = sin cos tan (b) π π π π + = + = sin 12 1 sec 12 sin 12 cos 12 1 2 2 2 2 ↑ ↑ θ θ = cos 1 sec θ θ + = sin cos 1 2 2 Now Work PROBLEM 7 9 5 Find the Exact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle Many problems require finding the exact values of the remaining trigonometric functions when the value of one of them is known and the quadrant in which θ lies can be found.There are two approaches to solving such problems. One approach uses a circle of radius r ; the other uses identities. When using identities, sometimes a rearrangement is required. For example, the Pythagorean identity θ θ + = sin cos 1 2 2 can be solved for θ sin in terms of θ cos (or vice versa) as follows: θ θ θ θ = − = ± − sin 1 cos sin 1 cos 2 2 2 where the + sign is used if θ > sin 0 and the − sign is used if θ < sin 0. Similarly, in θ θ + = tan 1 sec , 2 2 we can solve for θ tan (or θ sec ), and in θ θ + = cot 1 csc , 2 2 we can solve for θ cot (or θ csc ).
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