Ch 4.7: Inverse Trig Functions
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Ch 4.7: Inverse Trig Functions
Inverse of Sine
• Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one– Interval: , then y = sin(x) has an inverse– Written: y = arcsin(x) or y = sin-1(x)– Remember: y = sin-1(x) iff x = sin(y)
2 2x
y = sin(x)
x
y
2
2
Ex: Find the exact value for1
arcsin2
Ask yourself, where on the unit circle does sin = ?
Remember, you must be between
1
2
2 2x
:6
ANSWER
y = Arcsin (x)
Inverse of Cosine
• Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one– Interval: , then y = cos(x) has an inverse– Written: y = arccos(x) or y = cos-1(x)– Remember: y = cos-1(x) iff x = cos(y)
0 x
Ex: Find the exact value for1
arccos2
Ask yourself, where on the unit circle does cos = ?
Remember, you must be between
1
2
0 x :
3ANSWER
0
y = cos(x)
x
y
Inverse of Tangent
• Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one– Interval: , then y = tan(x) has an inverse– Written: y = arctan(x) or y = tan-1(x)– Remember: y = tan-1(x) iff x = tan(y)
2 2x
Ex: Find the exact value for arctan 1
Ask yourself, where on the unit circle does tan = 1?
Remember, you must be between 2 2
x
:
4ANSWER
2
2
y = tan(x)
x
y
y = Arctan (x)
Approximating Values
• By definition, inverses are supposed to be in radians• Check to see whether the number is in radians or
degrees• To do inverse trig functions, hit “2nd” then the trig
function• Round 4 places• Some will not work!
Ex: Arcsin(0.2447) = Ex: sin-1(2) =
0.2472
Inverse Prop.: Recall f(f-1(x))=x & f-1(f(x))=x
For -1 x 1 and sin(sin-1(x)) = x & sin-1(sin(y)) = y
For -1 x 1 and
cos(cos-1(x)) = x & cos-1(cos(y)) = y
For x is a real number and tan(tan-1(x)) = x & tan-1(tan(y)) = y
**Pay attention to make sure the values fall within the parameters of the inverse!**
2 2y
0 y
2 2y
1 1: sin sin
3Ex
1sin sin 1 1x x if x
!TRUE
1
3
: tan arctan 5Ex
1tan tan is realx x if x
!TRUE
5
5:arcsin sin
3Ex
arcsin(sin( ))
2 2y y if y
! So work each piece at timeFalse
5 3sin
3 2
3arcsin
2
3
More complex problems• Determine the quadrant• Draw a triangle, label the parts• Using the triangle, answer the problem
1 3: cos sin
5Ex
arcsin :2 2
x
Thus, either Quadrant I or IV. Since -3/5, you are in IV!!
cosA
H
4
5
1 3: sin tan
2Ex
1tan :2 2
x
Thus, either Quadrant I or IV. Since 3/2, you are in I!!
13sin
O
H
3
13 3 13
13
Most complex problems• Follow the same rules from previous slide, but
now you will have variables in your answer
1 4: cot cosEx
x
2 16x
2 2 2
:
4
Pythagorean Theorem
opposite x 2 2 16opposite x
2 16opposite x
cota
o
2
4
16x
2
2
16
16
x
x
2
2
4 16
16
x
x
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