DAY 5
Chapter 6.1 and 6.2 Inverse Trig Functions
Graph
Inverse
y = sin(x)
y = sin-1(x)
y = cos(x)
y = cos-1(x)
y = tan(x)
y = tan-1(x)
Inverse Trig Functions
Domain
Range
sin−1 x
⎡ −1,1⎤
⎣
⎦
⎡ π π⎤
⎢− 2 , 2 ⎥
⎣
⎦
cos−1 x
⎡ −1,1⎤
⎣
⎦
⎡0, π ⎤
⎣
⎦
tan−1 x
(−∞,∞)
⎛ π π⎞
⎜⎝ − 2 , 2 ⎟⎠
Inverse Reciprocal Trig Functions
Domain
Range
csc−1 x
(−∞,−1]∪[1,∞)
⎡ π ⎞ ⎛ π⎤
⎢ − ,0⎟ ∪ ⎜ 0, ⎥
⎢⎣ 2 ⎠ ⎝ 2 ⎥⎦
sec−1 x
(−∞,−1]∪[1,∞)
⎡ π ⎞ ⎛π
⎤
⎢0, ⎟ ∪ ⎜ ,π ⎥
⎢⎣ 2 ⎠ ⎝ 2
⎥⎦
cot−1 x
(−∞,∞)
(0,π )
1. Inverse Regular Functions: sin/cos/tan
1.
Find the angle that has a
sin/cos/tan of the ratio given.
2.
Make sure the angle is in the
correct range (output).
Example 1: sin–1 (½) =
Inverse sin range: ___________
Inverse cos range: ___________
Inverse tan range: ___________
⎛
Example 2: cos–1 ⎜ −
⎝
2⎞
⎟ =
2 ⎠
Example 3: arctan (0) =
2. Inverse Reciprocal Functions: csc/sec/cot
1.
Convert the reciprocal to the
regular trig function and
reciprocate the ratio.
2.
Follow steps #1 and #2 above.
3.
Check that the answer is in the
correct range (output) for the
RECIPROCAL FUNCTION (if it is not in
the correct range, it is undefined).
Inverse csc range: ___________
Inverse sec range: ___________
Inverse cot range: ___________
⎛ 2 3⎞
⎟ =
3 ⎠
⎝
Example 1: csc–1 ⎜ −
Example 2: arcsec (1) =
Example 3: cot-1 ( − 3 ) =
3. Inverse on the inside: sin(sin-1(x))
1.
If x is in the correct domain for
inverse sine, the answer is x.
Example 1: sin(sin-1(1)) =
Inverse sin domain: ___________
Inverse cos domain: ___________
Inverse tan domain: ___________
Example 2: cos (cos-1(0.7)) =
Example 3: tan (tan-1(9)) =
2.
If x is not in the correct domain,
the answer is undefined.
Example 4: sin(sin-1(9)) =
Example 5: cos(cos-1(–2)) =
Example 6: tan(tan-1(–2)) =
4. Regular Function on the Inside: sin-1(sin(x))
1.
If x is in the restricted domain of the
inside function (range of the inverse
function), the answer is x.
sin-1(sin(x)) = x,
−
π
3
Example 1: sin-1(sin( − )) =
π
π
≤x ≤
2
2
Example 2: cos-1(cos(
cos-1(cos(x)) = x,
0 ≤ x ≤π
tan-1(tan(x)) = x, −
π
<x
2
<π
2
2.
If x is not in the restricted domain of
the inside function…
3.
find the other angle (the one with
the same sin/cos/tan in the correct
domain)
5π
)) =
6
Example 3: tan-1(tan(0)) =
Example 4: sin-1(sin(
5π
)) =
6
π
2
Example 5: cos-1(cos( − )) =
OR:
3.
Evaluate the inside.
4.
Then evaluate the outside function
and use the correct range of the
inverse.
Example 6: tan-1(tan(
2π
)) =
3
5. Combinations of Trig Functions (similar to #3 and 4)
1.
Inverse function on the inside:
Draw and label the triangle.
Use the triangle to evaluate the outside function (in the correct range).
If it is not in the domain of the outside function, it’s undefined.
Example 1: cos(sin-1(–1/2))
2.
Example 2: sec(tan-1( − 3 ))
Regular function on the inside:
Evaluate the inside function.
Evaluate the outside function (in the correct range).
Example 3: sin-1(cos(
5π
))
4
Example 4: cos-1(sin(
4π
))
3