Inverse Trigonometric Ratios
Inverse trigonometric ratios are the trigonometric ratios that are used to find the value of the unknown angle with a given measure of the ratio of sides of the right-angled triangle. As we have used angles to find the trigonometric ratios of the sides of the triangle, similarly we can use the trigonometric ratios to find the angle. For example sin(θ) = (Opposite)/(Hypotenuse), hence we can get angle as sin-1(Opposite)/(Hypotenuse)= θ.
Using inverse trigonometric ratios we have found out the inverse of the trigonometric function(sin-1x) which is not the same as 1/sinx. The inverse is indicative inverse and is not an exponent. We can find out the inverse trigonometric ratio of any trigonometric function. Let us learn about them in the following sections along with a few solved examples.
What are Inverse Trigonometric Ratios?
Inverse trigonometric ratios are the inverse of the trigonometric functions operating on the ratio of the sides of the triangle to find out the measure of the angles of the right-angled triangle. The inverse of a function is denoted by the superscript "-1" of the given trigonometric function. For example, the inverse of the cosine function will be cos-1. The inverse of the trigonometric function is also written as an "arc"-trigonometric function, for example, arcsin will be the inverse of the sine function. Inverse trigonometric ratios are used when we have the measure of the sides of the right-angled triangle and want to know the measure of the angles of the triangle. Let us discuss each inverse trigonometric ratio for the trigonometric functions one by one.
Inverse Sine or Arcsine
As we know, the sine of the angle gives the ratio of the opposite side to the angle and the hypotenuse of the triangle. Therefore the inverse of the sine of the ratio of the opposite side and the hypotenuse will give the respective angle. So we can write:
sin-1(Opposite)/(Hypotenuse)= θ
Where
- θ is the base angle of the right-angled triangle.
Inverse Cosine or Arccos
The cosine of the angle gives the ratio of the base of the triangle and the hypotenuse of the triangle, therefore the inverse of the cosine of the ratio of the base and the hypotenuse will give the respective angle. So we can write:
cos-1(Base)/(Hypotenuse) = θ
Where
- θ is the base angle of the right-angled triangle.
Inverse Tan or Arctan
As we know, the tan of the angle gives the ratio of the opposite side to the angle and the base of the triangle, therefore the inverse of the tan of the ratio of the opposite side and the base will give the respective angle. So we can write:
tan-1(Opposite)/(Base) = θ
Where
- θ is the base angle of the right-angled triangle.
Similarly we can find out inverse of other trigonometric ratios:-
- Cosec-1 (Hypotenuse)/(Opposite) = θ
- Sec-1 (Hypotenuse)/(Base) = θ
- Cot-1 (Base)/(Opposite) = θ
Inverse Trigonometric Ratios Table
The following table lists some examples of the sin−1 operation:
| Trigonometric Ratios | Inverse Trigonometric Ratios |
|---|---|
|
sin 0 = 0 |
sin−10 = 0 |
|
sin(π/6) =1/2 |
sin−1(1/2) = π/6 |
|
sin(π/4) = 1/√2 |
sin−1(1/√2) = π/4 |
|
sin(π/3) = √3/2 |
sin−1(√3/2) = π/3 |
|
sin(π/2) = 1 |
sin−11 = π/2 |
Here are some examples of the cos−1 operation:
|
cos0 = 1 |
cos−11 = 0 |
|
cos(π/6) = √3/2 |
cos−1(√3/2) = π/6 |
|
cos(π/4) =1/√2 |
cos−1(1/√2) = π/4 |
|
cos(π/3)=1/2 |
cos−1(1/2) = π/3 |
|
cos(π/2) = 0 |
cos−10 = π/2 |
And a few examples of the tan−1 operation:
|
tan 0 = 0 |
tan−1 0 = 0 |
|
tan(π/6) =1/√3 |
tan−1(1/√3) = π/6 |
|
tan(π/4) =1 |
tan−1(1) = π/4 |
|
tan(π/3) = √3 |
tan−1(√3) = π/3 |
Applications of Inverse Trigonometric Ratios
Inverse trigonometric ratios have wide usage in the field of engineering, construction, and architecture. Inverse trigonometric ratios are the easiest way to find the unknown angle, hence in the places wherever we want to know the angle for our help, we use Inverse trigonometric ratios and quickly get the desired output. A few of the applications of Inverse trigonometric ratios are given below:
- Used to find the measure of the unknown angles of a right-angled triangle.
- Used in measuring the angle of depth or angle of inclination.
- Architects use it to calculate the angle of a bridge and the supports.
- Used by carpenters to create a desired cut angle.
Inverse Trigonometric Ratios Formulas
These are the very few basic formulas of inverse trigonometric ratios but based upon the trigonometric functions we can obtain more inverse trigonometric formulas. A few of the inverse trigonometric ratio formulas related to the inverse trigonometric functions are shown below.
- Sin-1(-x) = -Sin-1x
- Tan-1(-x) = -Tan-1x
- Cosec-1(-x) = -Cosec-1x
- Cos-1(-x) = π - Cos-1x
- Sec-1(-x) = π - Sec-1x
- Cot-1(-x) = π - Cot-1x
- Sin-1x = Cosec-11/x
- Cos-1x = Sec-11/x
- Tan-1x = Cot-11/x
- Sin-1x + Cos-1x = π/2
- Tan-1x + Cot-1x = π/2
- Sec-1x + Cosec-1x = π/2
- Sin-1x + Sin-1y = Sin-1(x.√(1 - y2) + y√(1 -x2))
- Sin-1x - Sin-1y = Sin-1(x.√(1 - y2) - y√(1 -x2))
- Cos-1x + Cos-1y = Cos-1(xy - √(1 - x2).√(1 -y2))
- Cos-1x - Cos-1y = Cos-1(xy + √(1 - x2).√(1 -y2))
- Tan-1x + Tan-1y = Tan-1(x + y)/(1 - xy)
- Tan-1x - Tan-1y = Tan-1(x - y)/(1 + xy)
Important Notes:
Here are a few important notes related to inverse trigonometric ratios.
- The inverse of the trigonometric function is also written as an "arc"-trigonometric function.
- Inverse trigonometric ratios have wide usage in the field of engineering, construction, and architecture.
Related Topics:
Here are a few related topics related to inverse trigonometric ratios.
FAQs on Inverse Trigonometric Ratios
What are Inverse Trigonometric Ratios?
Inverse trigonometric ratios are the inverse of the trigonometric functions operating on the ratio of the sides of the triangle to find out the measure of the angles of the right-angled triangle. The inverse of a function is denoted by the superscript "-1" of the given trigonometric function. For example, the inverse of the cosine function will be cos-1. For example:
- sin-1(Opposite)/(Hypotenuse)= θ
How Do You Find Missing Angles Using Inverse Trigonometric Ratios?
As we have used angles to find the trigonometric ratios of the sides of the triangle, similarly we can use the trigonometric ratios to find the angle. For example sin(θ) = (Opposite)/(Hypotenuse), hence we can get angle as sin-1(Opposite)/(Hypotenuse)= θ. Further we can make use of the trigonometric tables to find the missing angle values.
What are the Six Inverse Trigonometric Ratios?
The six inverse trigonometric ratios are:
- sin-1(Opposite)/(Hypotenuse)= θ
- cos-1(Base)/(Hypotenuse) = θ
- tan-1(Opposite)/(Base) = θ
- Cosec-1 (Hypotenuse)/(Opposite) = θ
- Sec-1 (Hypotenuse)/(Base) = θ
- Cot-1 (Base)/(Opposite) = θ
What are the Applications of Inverse Trigonometric Ratios?
Inverse trigonometric ratios have wide usage in the field of engineering, construction, and architecture. Inverse trigonometric ratios are the easiest way to find the unknown angle, hence in the places wherever we want to know the angle for our help, we use Inverse trigonometric ratios and quickly get the desired output. A few of the applications of Inverse trigonometric ratios are given below:
- Used to find the measure of the unknown angles of a right-angled triangle.
- Used in measuring the angle of depth or angle of inclination.
- Architects use it to calculate the angle of a bridge and the supports.
- Used by carpenters to create a desired cut angle.
How Do You Find Inverse Trigonometric Function?
The inverse trigonometric functions of inverse sine, inverse cosine, or inverse tangent can be found from the basic trigonometric ratios.
Sin θ = x and θ = Sin−1x
What Are Arcsine, Arccosine, and Arctangent?
The terms arcsine, arccosine, and arctangent are the inverse ratio of the trigonometric ratios Sinθ, Cosθ, and Tanθ.
- θ = sin-1x
- θ = cos-1y
- θ= tan-1z