This lesson explains the basics and relationships of Inverse tangent
and core Inverse Funtions Arcsine, Arccosine and Arctangent. The six core functions are: sin^{1}, cos^{1}, tan^{1}, csc^{1}, sec^{1}, and cot^{1}. These are also written as arcsin, arcos, arctan, arccsc, arcsec, arccot. The presentation will be done by the instructor in own handwriting and explanation with the help of several examples with video watching. The learning will help in doing home work and solving the real life application problems.
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The inverse functions tell you which angle would produce the value you've specified with the corresponding trigonometric function. E.g. arcsine ()
gives you the angle whose sine equals .
The basic idea of the inverse function of tangent is to find tan^{1} 1, in simple language would mean ‘what angle has tangent equal to 1?’ Think about it based on earlier learning and the answer is 45.
Thus it may be said that tan^{1}1 = 45.
When using for radians, this would mean
tan^{1} 1 = /4.
Notice that
the range
of tan^{1} is restricted to (–90,
90)
or (/2,
/2).
Now the earlier learning may be reviewed i.e. the tangent (tan) is defined as the ratio of the opposite side to the adjacent side in a right triangle. The fundamental relationships of sin, cos and tan are useful for finding either a side or an angle. If the requirement is to find the angle, you need an arcsine, arrcosine, or arctangent, also denoted arcsin, arccos, arctan. Notice that tan^{1}
here means inverse tan or arctan, not ‘tan to the power of negative one’, and also as such you may not see the connection between the geometric definition and the trigonometric definition. To better understand, here is an example:
tan
= opp/adj
arctan(tan )
= arctan(opp/adj)
= arctan(opp/adj)
Another notation of inverse trig function that may avoid confusion is:
tan^{1} ()
= arctan ()
This applies to other trigonometry functions as well.
To evaluate inverse trig function notice that the following equivalent statement:
= tan^{1} ()
()
= tan ()
= tan^{1} ()
Definition of tan functions:
For y = f(x) = tan^{1} (),
D (f) = R, R (f) = (/2,
/2)
For y = f(x) = cot^{1} (),
D (f) = R, R (f) = (0, )
Important properties :
Look at some of the problems get better understanding how it works:
Remember the related Theorems, it will easy your working the solutions and doing home work.
• y = arcsin x,
sin y = x , with 1 < x < 1 and 
/
2 <= y <=
/
2
• y = arcos x
cos y = x, with 1 < x < 1 and 0 <= y
<=
• y = arctan x
tan y = x, with 
/ 2 < y <
/ 2
Example 1: Given find the exact value of arctan( 1 )
Step 1: arctan(1)( Given).
Step 2: Recall the relevant theorem from above given,
y = arctan x tan y = x,
with 
/2
< y <
/2
i.e.
tan y =  1 with 
/2
< y <
/2
Step 3: Refer the trigonometric table of special angles. You’ll find tan (/4)
= 1
Step 4: Recall that tan (x) =  tan x.
Thus tan (/4)
= 1
Step 5: Also tan y = 1.
Thus y = 1 and therefore y = ( /4 )
Value of arctan(1 ) is ( /4 ), as the final answer.
Example 2: Given Evaluate tan(tan^{1}(3))
Step 1: You may recall: tan [tan^{1}()]
=
,
and tan^{1}[tan ()]
=
Step 2: Using this fact, you can say that equals to 3, as the final answer.
Example 3: Given simplify the trigonometric expression: csc ( arctan
)
Step 1: Say x = csc ( arctan
) and y = arctan
so that x = csc y = 1 / sin y.
Step 2: Using the relevant theorem from learning above,
y = arctan
may also be written as tan y =
with 
/
2 < y <
/
2
Step 3: You know that: tan^{2}y = sin^{2}y / cos^{2}y = sin^{2}y / (1  sin^{2}y)
(From trigonometric functions relationship)
Step 4: Solving above for sin y, gives sin y = [tan^{2}y / (1 + tan^{2}y) ]
=
(tan y)/ [(1
+ tan^{2}y)]
Step 5: As for /2
< y <= 0;sin y is negative and tan y is also negative, thus (tan y) = tan y and sin y = (tan y) / /2
[(1
+ tan^{2}y)]
= tan y /[
(1 + tan^{2}y)].
Step 6: For 0 < = y < /
2; sin y is positive and tan y is also positive, thus
(tan y) = tan y
and
sin y = tan y /[
(1 + tan^{2}y) ] Now
x = csc ( arctan
) = 1 / sin y = [(1
+
^{2})]
/
,
as the final answer.
Remember:
• arctan refers to "arc tangent", or the radian measure of the arc on a circle corresponding to a given value of tangent.
• it is important here to note that in tan^{1} , the “1” is not an exponent and therefore tan^{1}
1/ (tan ).
In fact tan^{1} here means inverse tan or arctan, not ‘tan to the power of negative one’.
• the trigonometric functions are periodic, many different angles may give the same values. E.g. 0
and 360
both have the same sine i.e. (0). The inverse functions will always return the angle closest to 0
that would produce the specified trigonometric value.
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