4.
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Again, Mathcad will only give answers that lie in the range of the inverse sine function, i. e.,
Check using Mathcad
==>
==>
Divide by 2
or
==>
Use unit circle
Take sin(2x) as a common factor
Take as a common factor
Thus,
or
Use the zero product property
or
==>
or
Solve for t
or
Substitute cos(x) for t
==>
or
==>
Given
To find the other angle (in Quadrant IV) whose cosine is 1/3, we subtract 70.5 from 360:
Given that cos-1(1/3) = 70.5o , solve for x in the interval
Solution:
Let
Substitute t for cos(x)
==>
Factor
or
(Which property did we use?)
or
The solutions are:
If we substitute x = 180o in the original equation, we get . Similarly for the value
x = 270o. Thus, x = 180o and x = 270o are not solutions.
However, because we squared both sides in the process of the solution, one has to be careful about introducing "extraneous" solutions (solutions that will not satisfy the original equation).
or
or
or
Use zero product property
or
Divide by 2
Math 002-041
Major Quiz 8 Solution
Nov. 29, 2004
Answer all questions. Show all your work
Solve the following problems for x:
1.
Solution:
Square both sides
Expand
Use the identity
Using the fact that the sine function is odd
The sine function is negative in Quadrant III and IV. Using the method of the unit circle (see Lecture Notes web page), we have
or
Check using Mathcad:
Remark
Mathcad returned only one value because it used the inverse sine function (sin-1 ). Recall that the range of sin-1(x) is between and .
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3.
Solution:
Group to take common factors
or
Remark

Mathcad can be used to check your answer using the "solve" command as follows:
The solution is given in radians which is the same as the one we obtained in degrees.
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2.
Solution:
Using
Where
and