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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
β€’ To have an inverse, a function must be one to one and
have a reflection across the line y = x.
β€’ However, none of the 6 trig functions have an inverse
because they are periodic and not one to one.
β€’ In order for the trig functions to have an inverse, the
domains and ranges must be restricted.
β€’ Take a look at the Inverse Trig worksheet for the Domain
and Range of each function.
Restrictions for an Inverse
sin 𝑦 = π‘₯
𝑖𝑓𝑓
𝑦 = π‘Žπ‘Ÿπ‘ sin π‘₯
𝑦 = 𝑠𝑖𝑛
or
βˆ’1
y = π‘ π‘–π‘›βˆ’1 π‘₯
1
π‘₯β‰ 
sin π‘₯
Example
1
𝑦 = π‘Žπ‘Ÿπ‘ sin βˆ’
2
1
sin 𝑦 = βˆ’
2
Sine of what angle is
1
equal to βˆ’ ?
2
πœ‹
𝑦=βˆ’
6
Example
𝑦 = π‘Žπ‘Ÿπ‘ cos 0
cos 𝑦 = 0
Cosine of what angle is equal to 0?
πœ‹
𝑦=
2
Example
𝑦 = π‘Žπ‘Ÿπ‘ tan 3
π‘‘π‘Žπ‘› 𝑦 = 3
sin 𝑦
3
=
cos 𝑦
1
3
sin 𝑦
= 2
1
cos 𝑦
2
πœ‹
𝑦=
3
Example with Calculator
𝑦 = π‘Žπ‘Ÿπ‘ sin(0.3)
MODE must be in RADIANS
𝑦 = π‘ π‘–π‘›βˆ’1 (0.3)
𝑦 β‰ˆ 0.305
Example with Calculator
𝑦 = π‘Žπ‘Ÿπ‘ sin(πœ‹)
MODE must be in RADIANS
𝑦 = π‘ π‘–π‘›βˆ’1 (πœ‹)
𝑦 β‰ˆ 𝐸𝑅𝑅𝑂𝑅
WHY?
πœ‹
πœ‹
βˆ’ β‰€πœƒβ‰€
2
2
SOH – CAH – TOA
β€’ Given: 𝑦 = π‘Žπ‘Ÿπ‘ sin π‘₯
β€’ Find: cos 𝑦
𝑦 = π‘Žπ‘Ÿπ‘ sin π‘₯
π‘₯ 𝑂𝑝𝑝
π‘–π‘šπ‘π‘™π‘–π‘’π‘  sin 𝑦 = =
1 𝐻𝑦𝑝
β€’ cos 𝑦 = cos(π‘Žπ‘Ÿπ‘ sin π‘₯)
β€’ Use Right Triangle Trig
to Solve
β€’ cos 𝑦 =
β€’ cos 𝑦 =
𝐴𝑑𝑗
𝐻𝑦𝑝
1βˆ’π‘₯ 2
1
Opp = x
y
Adj = b
π‘₯ 2 + 𝑏2 = 12
𝑏2 = 1 βˆ’ π‘₯ 2
𝑏 = ± 1 βˆ’ π‘₯2
𝑏 = 1 βˆ’ π‘₯2
sin 𝑦 𝑖𝑠 π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’ π‘ π‘œ + 1 βˆ’ π‘₯ 2
SOH – CAH – TOA
β€’ Given: 𝑦 = π‘Žπ‘Ÿπ‘ s𝑒𝑐
5
2
β€’ Find: tan 𝑦
tan 𝑦 = tan π‘Žπ‘Ÿπ‘ sec
5
2
β€’ Use Right Triangle Trig to
Solve
β€’ tan 𝑦 =
β€’ tan 𝑦 =
𝑂𝑝𝑝
𝐴𝑑𝑗
1
2
5
𝑦 = π‘Žπ‘Ÿπ‘ s𝑒𝑐
2
5 𝐻𝑦𝑝
π‘–π‘šπ‘π‘™π‘–π‘’π‘  s𝑒𝑐 𝑦 =
=
2
𝐴𝑑𝑗
Opp = b
y
Adj = 2
22
+ 𝑏2
2
= 5
4 + 𝑏2 = 5
𝑏2 = 1
𝑏 = ±1
s𝑒𝑐 𝑦 𝑖𝑠 π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’ π‘ π‘œ 𝑏 = +1
Formative Assessment
β€’ Pg. 377 (5-19) odd