1. Group Thinking – CIC Problem
How much money was spent at lunch yesterday?

2. Section 5.1, Day 2
AP Calculus AB

3. Learning Targets Define Inverse Trig Functions
Evaluate Inverse Trig Functions
Properties of Inverse Trig Functions
Solve Inverse Trig Equations
Derivatives of Inverse Trig Equations

4. Inverse Trig Functions: Definitions
Domain
Range
y = arcsin(x)
y = sin-1(x)
−1 ≤𝑥≤1
− 𝜋 2 ≤𝑦≤ 𝜋 2
y = arccos(x)
y = cos-1(x)
0≤𝑦≤𝜋
y = arctan(x)
y = tan-1(x)
−∞ ≤𝑥≤∞
y = arccot(x)
y = cot-1(x)
y = arcsec(x)
y = sec-1(x)
𝑥 ≥1
0≤𝑦≤𝜋, 𝑦 ≠ 𝜋 2
y = arccsc(x)
y = csc-1(x)
− 𝜋 2 ≤𝑦≤ 𝜋 2 , 𝑦 ≠0

5. Evaluating Inverse Trig Functions
Find sin −1 − 1 2
=− 𝜋 6
Find cos −1 0
= 𝜋 2
Find tan −1 3
= 𝜋 3
Find sin −1 (0.3)
=0.305

6. Properties of Inverse Trig Functions
sin arcsin 𝑥 =x
tan arctan 𝑥 =𝑥
cos arccos 𝑥 =𝑥
arcsin sin 𝑦 =𝑦
arccos cos 𝑦 =𝑦
arctan tan 𝑦 =𝑦
These properties hold for the other trig functions as well

7. Solving Trig Equations: Example 1
Solve arctan (2𝑥 −3) = 𝜋 4
2𝑥−3= tan 𝜋 4
2𝑥−3=1
𝑥=2

8. Example 2 Given 𝑦= arcsin 𝑥 where 0<𝑦< 𝜋 2 , find cos 𝑦
Draw the triangle knowing sin 𝑦 = 𝑥 1
Then solve for the 3rd side using Pythagorean theorem: 1− 𝑥 2
Thus cos 𝑦 = 1− 𝑥 2 1

9. Derivatives of Inverse Trig Functions
𝑑 𝑑𝑥 sin −1 u = u ′ 1− 𝑢 2
𝑑 𝑑𝑥 cos −1 u = − u ′ 1− 𝑢 2
𝑑 𝑑𝑥 tan −1 u = u ′ 1+ 𝑢 2

10. Derivatives of Inverse Trig Functions
𝑑 𝑑𝑥 csc −1 𝑢 = − 𝑢 ′ 𝑢 𝑢 2 −1
𝑑 𝑑𝑥 sec −1 𝑢 = 𝑢 ′ 𝑢 𝑢 2 −1
𝑑 𝑑𝑥 cot −1 𝑢 =− 𝑢 ′ 1+ 𝑢 2

11. Example 3
𝑑 𝑑𝑥 arcsin 2𝑥
𝑢 ′ 1− 𝑢 2 = 2 1−4 𝑥 2

12. Example 4
𝑑 𝑑𝑥 [ arctan 3𝑥 ]
𝑢 ′ 1+ 𝑢 2 = 𝑥 2

13. Example 5
𝑑 𝑑𝑥 [ arctan 𝑒 𝑥 ]
𝑢 ′ 1+ 𝑢 2 = 𝑒 𝑥 1+ 𝑒 2𝑥

14. Example 6
𝑑 𝑑𝑥 arccos 𝑒 2𝑥
− 𝑢 ′ 1− 𝑢 2 =− 2 𝑒 2𝑥 1− 𝑒 4𝑥

15. Example 7 𝑑 𝑑𝑥 𝑥 2 sin −1 𝑥 𝑥 2 1 1− 𝑥 2 + sin −1 𝑥 𝑥 2

16. Example 8 𝑑 𝑑𝑥 𝑥 ta n −1 ln 𝑥 + 3𝑥−1 4
𝑥 1 𝑥 ln 𝑥 tan −1 ( ln 𝑥 ) +4 3𝑥−1 3 (3)
= ln 𝑥 tan −1 ( ln 𝑥 ) 𝑥−1 3

17. Summary of Derivatives
Power Rule
Constant Rule
Product Rule
Quotient Rule
Chain Rule
Trigonometric Rules
Implicit Differentiation
Exponential Differentiation
Logarithmic Differentiation
Inverse Trigonometric Differentiation