9-15-14
T2.1 e To Find the
Inverse Functions for
sin Ө, cos Ө, & tan Ө
“It’s an obstacle illusion”
–Alan-Edward Warren, Sr. 2014
1
Active Learning Assignment Questions?
2
LESSON: To find the inverse for sin Ө , cos Ө , and tan Ө.
This is written as Sin-1, Cos-1, and Tan-1. It can also be
written as Arcsin, Arccos, and Arctan. These two terms
are INTERCHANGEABLE!!!!!!!!
If sin Ө = ratio, and we know the angle, we can find the
ratio. (Put your calculator back in degrees.)
Ex: sin 30° = x Just use your calculator (or build
the triangle to find it).
0.5 = x
2
1
30
3
3
BUT, what if we know the ratio, but want to solve
for the angle (in degrees)? Start with:
sin   0.5
How do we get Ө by itself?
Sin sin   Sin 0.5
1
1
(Pronounced sine inverse)
  Sin 0.5
1
We take the sine
inverse of both sides!
Since Sin-1 inverses sin, this
is all we have left.
The same!
  30
Now, we use the calculator
(see 2nd function above sin).
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sin 
Sin 1 ratio
Ө
ratio
Ө
cos
ratio
Ө
tan 
ratio
Ө
Inverse functions
must pass both the
ratio
vertical line test
and the horizontal
line test because
Cos1 ratio
Ө
when we take the
inverse, we switch
ratio
the domain and
range (the ratio
Tan 1 ratio
becomes the
Ө
questions (x) and
the angle becomes
ratio
the answer (y)) 5
*Positive and Negative Quadrants for Inverse Functions
1
What is Cos
? = 60°
2
1
II
1
What is Cos  ?
2 = 120°
180°
1
1
Cos
( ratio) is a
*
III
positive angle in QII
*All positive in QI
90°
I
1
What is Sin
? = 30°
2
1
0°
1
1
What is Sin  ? = -30°
2
1
-90° *Sin (  ratio) and
IV
Tan 1 (  ratio) are
negative angles in QIV
*(Reciprocals go together, too.)
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Radian vs. Degree.
What is the difference between these two?
1
1
Sin 0.92 vs.
Sin 0.92
in degrees
in radians
You cannot tell by the expression. You must receive
instructions as to which one is needed. Once that is
established, just change your calculator to the
appropriate setting. Let’s try to find it in radians.
Sin 1 0.92
= 1.1681
In radians, to four decimal places.
Same as  sin   0.92 
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Try these:
Degrees, 2 dec. pl.
1
Sin 0.28
 sin   0.28
16.26°
Degrees, 2 dec. pl.
Cos  0.57 
 cos    0.57 
1
124.75°
Radians, 4 dec. pl.
4
Tan
3
1
4

 tan   
3

0.9273
Try:
Either Degrees
or Radians
Sin 1 1.2
Error!
Error!
Error!
Sin 1 
Ө
ratio
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What about Sec-1 x , Csc-1 x , and Cot-1 x ?
It’s the same process, except for changing the function
to it’s reciprocal. Given, find in degrees:
csc  1.5984
1
 1.5984
sin 
1
1
sin  
1.5984
1
1
Sin sin   Sin
1.5984
1
1
  Sin
1.5984
1
To solve for Ө , we use the
reciprocal property for csc Ө
Flip that equation!
Inverse both sides
1
  38.73
Simplify
The same!
Solve, in degrees,
two decimal places.
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Try:
Radians, 4 dec. pl.
Degrees, 2 dec. pl.
1. cot   2.9856
1
 2.9856
ta n 
1
tan  
2.9856
1
1
1
Tan ta n   Tan
2.9856
1
1
  Tan
2.9856
Ans:
2. sec  4.0345
1
 4.0345
cos 
1
cos 
4.0345
1
1
1
Cos cos  Cos
4.0345
1
1
  Cos
4.0345
2.  1.3203
1.  18.52
Ex : cot   ratio  Cot
1
ratio  Tan
1
1
ratio
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Active Learning Assignment:
P 289: Written Exercises 1 – 4
AND
Find each in degrees (2 dec. pl.).
(Make sure your calculator is in degrees)
1. cot Ө = 5.829
2. csc Ө = 3
3. sec Ө = (-1.8726)
4. sec Ө = 2.8
5. csc Ө = 8.29
6. cot Ө = 0.75
(Answers to these six are on the next page.)
TEST on T2.1a, T2.1c, T2.1d, T2.1e & T2.1f on
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Thursday, 9-18-14 (3 x 5 card, only)
Answers for 1 – 6 on previous page:
In Degrees:
1.
9.73°
2.
19.47°
3.
122.28°
4.
69.08°
5.
6.93°
6.
53.13°
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