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WARM UP A man on a 135-ft vertical cliff looks down at an angle of 16 degrees and sees his friend. How far away is the man from his friend? How far is the friend from the base of the cliff? LESSON 14 INVERSE FUNCTIONS Objective: To introduce inverse functions and use them to find the angles of right triangles. INVERSE FUNCTIONS Inverse functions sin-1 or arcsin cos-1 or arccos tan-1 or arctan They are used to find the missing angle in a right triangle. THESE ARE NOT THE SAME AS THE RECIPROCAL OF THE FUNCTION 1 Sin-1 x = sin x FINDING THE ANGLE To find the angle β– use one of the trig functions that you have the info for. 4 Ex: sin β = 5 β because we 5 3 don’t know the angle. α C 4 A Press [2nd] [sin-1] (4/5) This will give you the Degrees of angle β 3 cos 2 1 EXAMPLES 30o 3 cos 2 150o 1 sin 1 2 -30o tan 1 1 45o 1 WARM UP Find the angle – round to the nearest degree: Sin A = .9063 Cos B = .6428 Tan C = .4040 USING THE UNIT CIRCLE Inverse functions can be evaluated using the unit circle. Only the angles between -90 and 90 are used. 3 -1 Ex: sin ( ) could be either 60o or 2 o 120 , so we are taking the one that is less than 90o FIND THE ANGLES USING THE UNIT CIRCLE 3 sin 2 60o tan 1 1 45o 1 2 cos 2 1 45o COMBINING FUNCTIONS WITH INVERSES 2 means that you ) tan(cos 2 1 want to find the tangent of whatever angle has a cosine of 2 2 We are only going to do this on the calculator. 2 ) 1 tan(cos 2 1 PRACTICE Cos(tan-1(.95)) = .725 Sin(cos-1(5/6))= .5528 Tan(tan-1(2) = 2