Download 4.7

Chp 4.7
Inverse Trigonometry Functions (pg. 343)
Standard:
Algebra 2
Standard:
F.TF.2.1
Bellwork
Sketch the graph of the following functions.
1. y = –tan 2x
2. y = sec(3x – 𝜋) + 2
Objective:
To graph
inverse
trigonometric
functions.
Inverse Sine The inverse sine function is defined by:
Function
y = arcsin x if and only if sin y = x
Domain: [–1, 1]
𝜋 𝜋
Range: [− ,
2
2
]
Inverse Cosine The inverse cosine function is defined by:
Function
y = arccos x if and only if cos y = x
Domain: [–1, 1]
Range: [0, 𝜋]
Inverse The inverse tangent function is defined by:
Tangent
Function
y = arctan x if and only if tan y = x
Domain: (–∞, ∞)
𝜋 𝜋
Range: [− ,
2 2
]
Example 1 If possible, find the exact values.
a.
1
√3
b. 𝑠𝑖𝑛−1 2
𝑎𝑟𝑐𝑠𝑖𝑛 − 2
c. 𝑠𝑖𝑛−1 2
Example 2/3 Find the exact value.
a.
√2
𝑎𝑟𝑐𝑐𝑜𝑠 2
b. 𝑐𝑜𝑠 −1 (−1)
c. arctan 0
d. 𝑡𝑎𝑛−1 (−1)
Example 4 Use a calculator to approximate the value.
a. arctan(–8.45)
b. sin-1 0.2247
c. arccos 2
Inverse If –1 ≤ x ≤ 1 and − 𝜋≤ y ≤ 𝜋, then
2
2
Properties of
Trig. Function
sin(arcsin x) = x and arcsin(sin y) = y
If –1 ≤ x ≤ 1 and 0 ≤ y ≤ π, then
cos(arccos x) = x and arccos(cos y) = y
If x is a real number and
−
𝜋
2
𝜋
< y < , then
2
tan(arctan x) = x and arctan(tan y) = y.
Example 5 If possible, find the exact values.
a. tan[arctan(–5)]
5𝜋
b. arcsin⁡(𝑠𝑖𝑛 3 )
c. cos(cos-1𝜋)