683 7.1 Fundamental Identities Fundamental Identities Reciprocal Identities cot U = 1 tan U sec U = 1 cos U csc U = 1 sin U Quotient Identities tan U = sin U cos U cot U = cos U sin U Pythagorean Identities sin 2 U +cos 2 U =1 tan 2 U +1 =sec 2 U 1 +cot 2 U =csc 2 U Even-Odd Identities sin1−U2 = −sin U cos1−U2 =cos U tan1−U2 = −tan U csc1−U2 = −csc U sec1−U2 =sec U cot1−U2 = −cot U NOTE We will also use alternative forms of the fundamental identities. For example, two other forms of sin2 U +cos2 U =1 are sin2 U =1 −cos2 U and cos2 U =1 −sin2 U. Uses of the Fundamental Identities We can use these identities to find the values of other trigonometric functions from the value of a given trigonometric function. EXAMPLE 1 FindingTrigonometric Function Values Given One Value and the Quadrant If tan u = - 5 3 and u is in quadrant II, find each function value. (a) sec u (b) sin u (c) cot1-u2 SOLUTION (a) We use an identity that relates the tangent and secant functions. tan2 u + 1 = sec2 u Pythagorean identity a- 5 3b 2 + 1 = sec2 u tan u = - 5 3 25 9 + 1 = sec2 u Square - 5 3 . 34 9 = sec2 u Add; 1 = 9 9 -B34 9 = sec u Take the negative square root because u is in quadrant II. sec u = - 234 3 Simplify the radical: -334 9 = - 234 29 = - 234 3 , and rewrite. Choose the correct sign.
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