6.3: Inverse Trigonometric Functions • Graphical Approach: We learned that sine, cosine and tangent functions are periodic so they are not one-to-one. To define an inverse function for them, we restrict their domain to intervals that contains the largest one-to-one piece of their graph/ The following are the standard form of these restrictions. h π π i y = sin(x) restricted to domain − ; 2 2 y = arcsin(x) Domain: [−1; 1] x x h π π i Range: − ; 2 2 y = arccos(x) y = cos(x) restricted to domain [0; π] Domain: [−1; 1] x x Range: [0; π] y = tan(x) restricted π π to domain − ; 2 2 y = arctan(x) π y = 2 π y = − 2 Domain: (−∞; 1) π π π π x = − x = Range: − ; 2 2 2 2 • Inverse trigonometric functions and right triangles: sin(θ) = O=H arcsin(O=H) = θ cos(θ) = A=H arccos(A=H) = θ H O tan(θ) = O=A arctan(O=A) = θ θ A • Right triangles can be used to find sine, cosine and tangent of an inverse trig function: • sin (arcsin(x))= x 1 x p • cos (arcsin(x))= 1 − x2 θ p x 1 − x2 • tan (arcsin(x))= p 1 − x2 p • sin (arccos(x))= 1 − x2 1 p • cos (arccos(x))= x 1 − x2 p 1 − x2 θ • tan (arccos(x))= x x x p • sin (arctan(x))= p x2 + 1 2 x x + 1 1 • cos (arctan(x))= p θ 2 1 x + 1 • tan (arctan(x))= x 2 • Note that sin(arccos(x)) ≥ 0, cos(arcsin(x)) ≥ 0 and cos(arctan(x)) > 0 even if x is negative. this can be explained using the range of arc functions and quadrants of unit circle. • Also note that in the previous page if x < 0, the sign of two of compositions is negative and one of the compositions has a positive sign.This matches the quadrants that they belong to and therefore the found values work for non-acute angles as well. • Other notations for inverse functions: sin−1(x) = arcsin(x), cos−1(x) = arccos(x) and tan−1(x) = arctan(x). Note that the inverse function notation is only true for the restricted domain. • Warning: Even though sin(arcsin(x)) = x , cos(arccos(x)) = x and tan(arctan(x)) = x, the other way around for these compositions does not work. 3π 3π 5π 5π For example, arcsin sin 6= , arccos cos 6= and so on. 4 4 4 4 h π π i • However, if the domain is restricted to − ; , then arcsin(sin(x)) = x. 2 2 If the domain is restricted to [0; π], then arccos(cos(x)) = x. π π If the domain is restricted to − ; , then arctan(tan(x)) = x. 2 2 • When finding arc functions, it is very important to remember the following formulas: • Note that θ = arcsin(x)= ) x = sin(θ) • Note that θ = arccos(x)= ) x = cos(θ) • Note that θ = arctan(x)= ) x = tan(θ) • The steps in finding Composition: When finding trig function arc function x , follow these steps: 1. trig function arc function(x) =) trig function(θ)= x | {z } θ 2. Draw a right triangle with angle θ such that trig function of θ is x. Find the other trig functions. 3. Find trig function(θ). • Note the input of a trig function such as sine, cosine,... is an angle in radian or degree and the output is a ratio of two lengths. input(Measure of an Angle) Trig output( Ratio) • On the other hand, the input of an arc function is a ratio and the output is an angle. input(Ratio) Arc output( Measure of an Angle) 3 1. Write tan(sin−1(x)) as an algebraic expression in terms of x. x x (a) p (c) 1 − x2 1 − x 1 (b) (d)1 − x2 x 2. Find θ. 12 15 θ θ 5 9 3. A 20-ft ladder is leaning against a building. If the base of the ladder is 16 ft from the base of the building, what is the angle of elevation of the ladder? 20 ft θ 16 ft 4 4. Find each of the following. p !! −1 3 (d) tan(arctan(1:7)) 3 (a) tan cos (g) cos arcsin 5 2 (e) sin(arctan(1)) 4 −1 −1 (h) cos arcsin (b) tan tan 2 3 p !! 2 −3 −1 (f) cos arcsin (i) sin arccos (c) sin (4) 2 5 5. Evaluate the following. 7π 5π 7π (a) arccos cos (c) arccos cos (e) arcsin cos 6 6 6 7π 5π 7π (b) arcsin sin (d) arcsin sin (f) arccos sin 6 6 6 5 6. An observer views the space shuttle from a dis- tance of x = 10 mil from the launch pad. Ex- press the angle of elevation θ as a function of the height h of the space shuttle. h θ 10 6.