Trigonometry Lecture Notes_part1-1

• a maximum on the cosine curve represents a

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Unformatted text preview: • A maximum on the cosine curve represents a minimum on the secant curve. • A minimum on the cosine curve represents a maximum on the secant curve. Example 34 Graph 3sec 2 x y = - for 5 x π π- < < (Hint: use the graph of 3cos 2 x y = - ) Section 7.6 Inverse Trigonometric Functions Recall that in order for a function to have an inverse it must pass the horizontal line test (i.e.-it must be one-to-one). Also, the inverse function’s graph will be the reflection of the original function about the line y = x. Since the trig functions would fail to be one-to-one on their entire domain due to their periodic nature among other things, we must restrict their domains to be able to find their inverses. The Inverse Sine Function The inverse sine function , denoted by sin-1 , is the inverse of the restricted sine function: y = sin x , - π /2 < x < π / 2 . Thus, y = sin-1 x means sin y = x , “at what angle is the sine equal to x?” where - π /2 < y < π /2 and –1 < x < 1. We read y = sin-1 x as “ y equals the inverse sine at x .” We can graph the inverse of the sine function with the above restricted domain by simple reversing the points. For example, (x, y) is switched to (y, x). Finding Exact Values of sin-1 x • Let θ = sin-1 x . • Rewrite step 1 as sin θ = x . • Use the exact values from the unit circle to find the value of θ in [- π /2 , π /2] that satisfies sin θ = x . Example 35 Find the exact value of sin-1 (1/2) The Inverse Cosine Function The inverse cosine function , denoted by cos-1 , is the inverse of the restricted cosine function y = cos x , 0 < x < π . Thus, y = cos-1 x means cos y = x , where 0 < y < π and –1 < x < 1. Example 36 Find the exact value of cos-1 (- √ 3 /2) The Inverse Tangent Function The inverse tangent function , denoted by tan-1 , is the inverse of the restricted tangent function y = tan x , - π /2 < x < π /2 . Thus, y = tan-1 x means tan y = x , where - π /2 < y < π /2 and – ∞ < x < ∞ . Example 37 Find the exact value of 1 tan 3- Inverse Properties The Sine Function and Its Inverse sin (sin-1 x ) = x for every x in the interval [-1, 1]. sin-1 (sin x ) = x for every x in the interval [- π /2 , π /2 ]. The Cosine Function and Its Inverse cos (cos-1 x ) = x for every x in the interval [-1, 1]. cos-1 (cos x ) = x for every x in the interval [0, π ]. The Tangent Function and Its Inverse tan (tan-1 x ) = x for every real number x tan-1 (tan x ) = x for every x in the interval (- π /2 , π /2 ). Example 38 Find the exact values if possible of: a. ( ) 1 cos cos 0.6- b. 1 3 sin sin 2 π- c. ( ) 1 cos cos 2 π- Example 39 Find the exact value of 1 5 cos tan 12- . Example 40 Find the exact value of 1 1 cot sin 3-- . Example 41 Calculators and the Inverse Trigonometric Functions Use your calculator to find 1 1 sin 4- and ( ) 1 tan 9.65-- in radian mode. Graphs of the Three Basic Trigonometric Functions Arc tangent...
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