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: (−∞, ∞)

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 √ • cos (arcsin(x))= 1 − x2 θ √ x 1 − x2 • tan (arcsin(x))= √ 1 − x2

√ • sin (arccos(x))= 1 − x2 1 √ • cos (arccos(x))= x 1 − x2 √ 1 − x2 θ • tan (arccos(x))= x x

x √ • sin (arctan(x))= √ x2 + 1 2 x x + 1 1 • cos (arctan(x))= √ 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) √ (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. √ !! −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 √ !! 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