The Formula of Inverse Trigonometric Functions
Domain and Range of Inverse Trigonometric Formulas
Function |
Domain |
Range |
x |
[-1,1] |
[-[frac{pi}{2}], [frac{pi}{2}]] |
x |
[-1, 1] |
[0, [pi] ] |
x |
R |
[-[frac{pi}{2}], [frac{pi}{2}]] |
x |
R |
[0, [pi]] |
x |
R- [-1, 1] |
[0, [pi]] – {[frac{pi}{2}]} |
x |
R- [-1, 1] |
[-[frac{pi}{2}], [frac{pi}{2}]] – {0} |
It is important to note the following formulas considering the domain and range of inverse function
-
sin(sin-1x) = x, if -1 ≤ x ≤ 1 and sin-1(sin y) = y if -[frac{pi}{2}] ≤ y ≤ [frac{pi}{2}].
-
cos(cos-1x) = x, if -1 ≤ x ≤ 1 and cos-1(cos y) = y if 0 ≤ y(arccos) ≤ π.
-
tan(tan-1x) = x, if -∞ < x < ∞ and cos-1(cos y) = y if -[frac{pi}{2}] ≤ y(arctan) ≤ [frac{pi}{2}].
-
cot(cot-1x) = x, if -∞ < x < ∞ and cot-1(cot y) = y if 0 < y <π.
-
sec(sec-1x) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞ and sec-1(sec y) = y if -0 ≤ y ≤ π, y ≠ [frac{pi}{2}].
-
cosec(cosec-1x) = x, if -∞ ≤ x ≤- 1 or 1 ≤ x ≤ ∞ and cosec-1(cosec y) = y if -[frac{pi}{2}] ≤ y ≤ [frac{pi}{2}], y ≠ 0.
Inverse trigonometric functions are also known as ‘arc functions’ because, for a given value of the trigonometric function, they produce the length of arc needed to get the particular value.
Graph of Inverse Trigonometric Function
1 – Arcsine Function
inverse sine function is defined as
y = arcsin x for – [frac{pi}{2}] ≤ y ≤ [frac{pi}{2}]
y is the angle with sine x which means x = sin y
the graph of y = arcsin x
()
2 – Arccosine Function
The graph of cosine does not extend beyond the point you see in the graph (if it extended, there would be multiple values of y for each x value and we would no longer have a function). The start and endpoints are indicated with dots (-1,) and (1,0)
()
3 – Arctangent Function
This graph can extend beyond what you see in the positive and negative direction of x and it does not cross the dashed line.
The domain of arctan x is all values of x
The range for arctan x is – [frac{pi}{2}] < arctan x < [frac{pi}{2}]
()
4 – Arccotangent Function
The graph of arccotangent extends in the positive and negative x directions. As shown in the graph it does not stop at -8, 8
The domain of arccot x is all values of x
The range of arccot x is −2π < arccot x ≤ 2π (arccot x ≠ 0)
()
5 – Arcsecant Function
Here, in the graph of sec inverse x, the curve is defined outside of the portion between -1 and 1. The starting points (-1, π) and (1,0) with dots.
The domain of arcsec x is all values of x except −1 < x < 1
The range of arcsec x is 0 ≤ arcsec x ≤ π, arcsec x [neq frac{pi}{2}]
()
6 – Arccosecant Function
The graph extends from positive and negative x direction and is not defined between – 1 and 1
The domain of arccsc x is all values of x except – 1 < x < 1
The range
The range of arccsc x is – [frac{pi}{2}] ≤ arc csc x ≤ [frac{pi}{2}] , arccsc x [neq 0]
()
Solved Examples of Inverse Trigonometric Functions
1. Find the accurate value of each of the expression in [0, 2[pi]].
-
sin-1(−3[sqrt{2}])
-
cos-1(−2[sqrt{2}])
-
tan-1[sqrt{3}]
solution:
a. We get – 3 [sqrt{2}] from 30 – 60 – 90 triangle. Therefore, the reference angle for 3 [sqrt{2}] would be 60°. As it is sine it is negative and must be in the third or fourth quadrant. Here the answer is either 4 [frac{pi}{3}] or 5[frac{pi}{3}]
b. From the isosceles right triangle we get -2[sqrt{2}]. The reference angle will be 45° as it is cosine and negative. The angle is either on the second or third quadrant. The answer is 3 [frac{pi}{4}] or 5[frac{pi}{4}]
c. From the 30 – 60 – 90 triangle we get [sqrt{3}]. For the reference angle 60°, a tangent is [sqrt{3}]. In the first and third quadrant, the tangent is positive, therefore, the answer is [frac{pi}{3}] or 4 [frac{pi}{3}]
Note: Every example here has two answers which can be a problem when finding a single inverse for each trigonometric function. So, we have to restrict the domain in which inverse is found.
2. Get the value of (1.1106).
Solution: let B = (1.1106)
Then tan B = 1.1106
B = 48°
Tan 48 = 1.1106
Therefore, (1.1106 ) = 48°.