Download Other inverse trigonometric functions

Unit 8 Day 19 - Other Inverse
Trigonometric Functions
We will learn how to get the values of
the inverse trig functions csc-1, sec-1,
and cot-1
and
I will practice these skills.
February 27, 2017
Get a Calculator!
Inverse Functions
• Cos-1(x) – the angle whose cosine is x
– Domain [-1, 1]
– Range [0, ]
• Sin-1(x) – the angle whose sin is x
– Domain [-1, 1]
– Range [-/2, /2]
• Tan-1(x) – the angle whose tangent is x
– Domain (-,)
– Range [-/2, /2]
The Remaining Inverse Functions
• Sec-1(x) – the angle whose secant is x
– Domain (-, -1] and [1,)
– Range [0, ]
1
-1
-1
– sec (x) = cos  x 
• Csc-1(x) – the angle whose cosecant is x
– Domain (-, -1] and [1,)
– Range [-/2, /2]
– csc-1(x) = sin-1 x1 
• Cot-1(x) – the angle whose cotangent is x
– Domain (-,)
– Range (0, )
– Differences of opinion on range
Find the exact value of the following:
π
-1
-1
• csc 2 = sin (½)
6
• arccot 1
• csc-1(-1) =
• arcsec
2 3
3
π
4
1
-1

sin ( )
1
=
= arccos
sin-1(-1)
3
2

6
π

2
Find the exact value (work from the
inside out)
• csc(cos-1( -½ ))
2π
csc( 3 )
2 3
3
• sec[sin-1(0)]
sec(0)
1
Find the exact value (work from the
inside out)
•
7
-1

sec(sin ( 25 ))
sec 
r

x
25
24
 7 
sin  

 25 
1
y
7
sin   

25 r
y  7 r  25
x  24
Using the Calculator to get
Approximations
• Most calculators do not have keys for cot,
csc, and sec.
• The easiest way to evaluate them is to
convert them to their reciprocal.
• Example: sec-13
Put cos-1(1/3) in
the calculator
sec-13 ≈ 1.23
Evaluate to four decimal places
• Arcsec 10.9
1.4789
• Arccsc(-2.35)
-.4395
Using the Calculator to get
Approximations
• Cot-1 and tan-1 aren’t defined over the same
interval, so we can use cos-1 instead
• Example: cot-1-2
x
cot   2 
x2 + y2 = r2
0
y
(-2)2 + 12 = r2
x 2
4 + 1 = r2
cos   
r
5 = r2
5
cos-1(-2/√5)
So put
in the calculator
cot-1(-2) ≈ 2.68
√5 = r