pre cal worksheet
16
Math 1083 Worksheet 7 Getting Ready for Inverse Trigonometric Functions
Objectives: 1. Definition and notation of inverse functions
2. Graphs of inverse function
(1) Inverse functions
1. The inverse involves interchanging 𝑥 with 𝑦. The inverse of 𝑓(𝑥), denoted by 𝑓 −1(𝑥), will undo the operations of 𝑓 on the input; that is, 𝑓 −1(𝑓(𝑥)) = 𝑥 𝑎𝑛𝑑 𝑓(𝑓 −1(𝑥)) = 𝑥
2. The graphs of 𝑓(𝑥) and its inverse 𝑓 −1(𝑥) are symmetric over the line 𝑦 = 𝑥. If point (𝑎, 𝑏) is on the graph of 𝑓(𝑥), then point (𝑏, 𝑎) is on the graph of the inverse 𝑓 −1(𝑥).
#1. Let 𝑓(𝑥) = √𝑥. a) Verify that 𝑓 −1(𝑥) = 𝑥2, 𝑥 ≥ 0 b) Evaluate the following
𝑓(4) = 𝑓 −1(2) 𝑓 −1(4) = 𝑓(16)
b) Graph 𝑓(𝑥) and 𝑓 −1(𝑥) on the same xy-plane.
#2 Given the graph of 𝑓(𝑥). Draw the inverse 𝑓 −1(𝑥) of each graph.
a)
b)
17
c)
d)
e)
f)
#3 Give the angle and coordinates for each point in the unit circle below and use to find 𝜃.
For 0 ≤ 𝜽 ≤ 𝝅
1. cos 𝜃 = −√𝟑
𝟐 , 𝜃 =?
2. cos 𝜃 = √𝟐
𝟐 , 𝜃 =?
For −𝝅
𝟐 ≤ 𝜽 ≤
𝝅
𝟐
3. sin 𝜃 = 𝟏
𝟐 , 𝜃 =?
4. sin 𝜃 = −√𝟑
𝟐 , 𝜃 =?
18
For −𝝅
𝟐 ≤ 𝜽 ≤
𝝅
𝟐
5. sin 𝜃 = − 𝟏
𝟐 , 𝜃 =?
6. sin 𝜃 = 𝟎, 𝜃 =?
For 0 ≤ 𝜽 ≤ 𝝅
7. cos 𝜃 = −𝟏
𝟐 , 𝜃 =?
8. cos 𝜃 = 𝟎, 𝜃 =?
REVIEW:
x2 + y2 = r2, 𝐜𝐨𝐬 𝛉 = 𝒙
𝒓 , 𝐬𝐢𝐧𝜽 =
𝒚
𝒓
tan 𝜽 = 𝒚
𝒙 , csc 𝜽 =
𝒓
𝒚
cot 𝜽 = 𝒙
𝒚 , sec 𝜽 =
𝑯
𝑨
#4 Find the indicate values.
a) cos 𝜃 = 𝟐
𝟓 , 𝜃 𝑖𝑠 𝑖𝑛 𝑄𝑢𝑎𝑑. 𝐼
x = ___________, y = ___________, r = 5 tan 𝜃 = _____________
b) sin 𝜃 = −𝟓
𝟏𝟑 , 𝜃 𝑖𝑠 𝑖𝑛 𝑄𝑢𝑎𝑑. 𝐼𝑉
x = _____________, y = ___________, r = 13
cos 𝜃 = ____________
c) tan 𝜃 = 𝟓
𝟏𝟐 , 𝜃 𝑖𝑠 𝑖𝑛 𝑄𝑢𝑎𝑑. 𝐼𝐼𝐼
x = -12, y = ___________, r = ____________
cos 𝜃 = ____________
d) sin 𝜃 = 𝟒
𝟓 , 𝜃 𝑖𝑠 𝑖𝑛 𝑄𝑢𝑎𝑑. 𝐼𝐼
x = ______________, y = 4, r = ___________
tan 𝜃 = _____________