4.7 inverse trigonometric functions
DESCRIPTION
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) - PowerPoint PPT PresentationTRANSCRIPT
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
• Examplex y
1 -12
2 -13
3 -14
4 -15
x y
-12 1
-13 2
-14 3
-15 4
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 [- /2p , /2p ]
– 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 = arcsin iff siny = 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.a) arcsin(-1/2)
b) sin-1(√3/2)
c) sin-1 2
Graph Arcsine by Hand
xy
y = sin x
y = arcsin x
xy
Cosine
Is cosine one-to-one??
Arccosine
• Definition: – The inverse cosine function is defined by:y = arccosx iff cosy = 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 = arctanx iff tany = x
where -∞ < x < ∞ and –p/2 < y < /2p
• y = arctanx Domain: (-∞,∞)Range: (-
p/2, /2p )
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)