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```AP CALCULUS AB
5.6 Inverse Trigonometric Functions
Objectives: Develop properties of the six
inverse trig functions and differentiate an
inverse trig function
Inverse Functions
• To have an inverse, a function must be one to one and
have a reflection across the line y = x.
• However, none of the 6 trig functions have an inverse
because they are periodic and not one to one.
• In order for the trig functions to have an inverse, the
domains and ranges must be restricted.
• Take a look at the Inverse Trig worksheet for the Domain
and Range of each function.
Restrictions for an Inverse
sin 𝑦 = 𝑥
𝑖𝑓𝑓
𝑦 = 𝑎𝑟𝑐 sin 𝑥
𝑦 = 𝑠𝑖𝑛
or
−1
y = 𝑠𝑖𝑛−1 𝑥
1
𝑥≠
sin 𝑥
Example
1
𝑦 = 𝑎𝑟𝑐 sin −
2
1
sin 𝑦 = −
2
Sine of what angle is
1
equal to − ?
2
𝜋
𝑦=−
6
Example
𝑦 = 𝑎𝑟𝑐 cos 0
cos 𝑦 = 0
Cosine of what angle is equal to 0?
𝜋
𝑦=
2
Example
𝑦 = 𝑎𝑟𝑐 tan 3
𝑡𝑎𝑛 𝑦 = 3
sin 𝑦
3
=
cos 𝑦
1
3
sin 𝑦
= 2
1
cos 𝑦
2
𝜋
𝑦=
3
Example with Calculator
𝑦 = 𝑎𝑟𝑐 sin(0.3)
𝑦 = 𝑠𝑖𝑛−1 (0.3)
𝑦 ≈ 0.305
Example with Calculator
𝑦 = 𝑎𝑟𝑐 sin(𝜋)
𝑦 = 𝑠𝑖𝑛−1 (𝜋)
𝑦 ≈ 𝐸𝑅𝑅𝑂𝑅
WHY?
𝜋
𝜋
− ≤𝜃≤
2
2
SOH – CAH – TOA
• Given: 𝑦 = 𝑎𝑟𝑐 sin 𝑥
• Find: cos 𝑦
𝑦 = 𝑎𝑟𝑐 sin 𝑥
𝑥 𝑂𝑝𝑝
𝑖𝑚𝑝𝑙𝑖𝑒𝑠 sin 𝑦 = =
1 𝐻𝑦𝑝
• cos 𝑦 = cos(𝑎𝑟𝑐 sin 𝑥)
• Use Right Triangle Trig
to Solve
• cos 𝑦 =
• cos 𝑦 =
𝐴𝑑𝑗
𝐻𝑦𝑝
1−𝑥 2
1
Opp = x
y
𝑥 2 + 𝑏2 = 12
𝑏2 = 1 − 𝑥 2
𝑏 = ± 1 − 𝑥2
𝑏 = 1 − 𝑥2
sin 𝑦 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑠𝑜 + 1 − 𝑥 2
SOH – CAH – TOA
• Given: 𝑦 = 𝑎𝑟𝑐 s𝑒𝑐
5
2
• Find: tan 𝑦
tan 𝑦 = tan 𝑎𝑟𝑐 sec
5
2
• Use Right Triangle Trig to
Solve
• tan 𝑦 =
• tan 𝑦 =
𝑂𝑝𝑝
𝐴𝑑𝑗
1
2
5
𝑦 = 𝑎𝑟𝑐 s𝑒𝑐
2
5 𝐻𝑦𝑝
𝑖𝑚𝑝𝑙𝑖𝑒𝑠 s𝑒𝑐 𝑦 =
=
2
𝐴𝑑𝑗
Opp = b
y