4.7 Inverse Trigonometric Functions

4.7 Inverse Trigonometric Functions *Intro to Inverse Functions *One-to-one *Inverse of Sine

Inverse Functions • f-1 (x) is the inverse of f(x) • If and only if the domain of f(x) is equal to the range of f-1(x) AND the range of f(x) is equal to the domain of f-1(x) • In English the x and y’s switch • Example f(x)= f-1(x)=

Do all functions have inverses? • Linear Functions • Quadratic Functions

Cubic Functions • Absolute Value

One-to-One • A function must be one-to-one to have an inverse • One-to-one: there is only one x for each y • Horizontal Line Test • Solve for y • You can write a restriction on the domain to make a function one-to-one

Trig Functions – Sine Is y = sinx one-to-one??

Inverse of Sine • f(x) = sinx has an inverse over the interval [-p/2,p/2] • sin x is increasing over the interval • takes on the full range values -1 < y < 1 • sine is one-to-one

Inverse of Sine • Inverse of sine is called: • inverse sine: sin-1 • arcsine: arcsin • the angle (or arc) whose sine is x • Sine: input = angle, output = ratio of the sides • Arcsine: input = ratio of sides, output = angle

Common Mistake • Don’t be FOOLED!!! • sin-1 is not the same as 1/sin

Arcsine • Definition: • The inverse sine function is defined by: y = arcsiniffsiny = x where -1 < x < 1 and –p/2 < y <p/2 • y = arcsin Domain: [-1,1] Range: [–p/2, p/2]

Evaluating the Inverse Sine Function • If possible, find the exact value. • arcsin(-1/2) • sin-1(√3/2) • sin-12

Graph Arcsine by Hand y = sin x y = arcsin x

Cosine Is cosine one-to-one??

Arccosine • Definition: • The inverse cosine function is defined by: y = arccosxiffcosy= x where -1 < x < 1 and 0 <y <p • y = arccosx Domain: [-1,1] Range: [0, p]

Graph of Arccosine

Evaluating Arccos • arccos(1/√2) • cos-1 (-1) • arccos(0)

Tangent

Arctangent • Definition: • The inverse tangent function is defined by: y = arctanxifftany= x where -∞ < x < ∞ and –p/2 < y < p/2 • y = arctanxDomain: (-∞,∞) Range: (-p/2, p/2)

Graph of Arctan

Evaluating Inverse Tangent • arctan 0 • tan-1 (-1) • arctan(√3)

Evaluating on the Calculator • Remember!!! Your input is the ratio of the sides and your output is the angle • The mode is the units your answer (angle) will be in • An error message is most likely because you are entering a number that is not in the domain

Evaluating Compositions of Functions • tan(arccos2/3) • cos(arcsin(-3/5)) • sin(arctan2)

4.7 Inverse Trigonometric Functions