Inverse Trigonometric Functions – CBSE Class 12 Maths
Inverse Trigonometric Functions, in CBSE Class 12 Maths, are basically the inverse functions of trigonometric functions. In particular, they are usually the;
1. Inverse of Sine Functions,
2. Inverse of Cosine,
3. Inverse of Tangent,
4. Inverse of Cotangent,
5. Inverse of Secant, and
6. Inverse of Cosecant Functions
These Inverse Trigonometric Functions are used to derive an angle from the trigonometric ratio of any angle. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
The study of trigonometry was first started in India. The ancient Indian Mathematician, Arya Bhatta, Brahmagupta, Bhaskara I, and Bhaskara II got important results of trigonometry. All this knowledge went from India to Arabia and then from there to Europe. The Greeks have also started the study of trigonometry but their approach was clumsy that when the Indian approach became known, it was immediately adopted throughout the world. Bhaskara I gave formulae to find the values of sine functions for angles more than 900.
An exact expression for sines or cosines of 180, 360, 540, 720, etc., were given by Bhaskara II.
The symbols – sin-1 x, cos-1 x, tan-1 x, cosec-1 x, sec-1 x, and cot-1 x, for , etc., were suggested by the astronomer Sir John F.W. Hersehel.
The trigonometric functions are real functions that relate the sides of a right triangle to its angles. Trigonometric functions are typically used to calculate unknown lengths or angles in a right triangle. For any right triangle, the six trigonometric ratios are there, viz.,
1. Sine (sin),
2. Cosine (cos),
3. Tangent (tan),
4. Cosecant (cosec),
5. Secant (sec), and
6. Cotangent (cot).
The trigonometric functions can be derived using the following formulae:
For any angle θ,
1. sin θ = side opposite to angle/Hypotenuse.
2. cos θ = Side adjacent to the angle/Hypotenuse
3. tan θ = side opposite to angle/ Side adjacent to the angle
4. cosec θ = Hypotenuse/ side opposite to angle
5. sec θ = Hypotenuse/ Side adjacent to the angle
6. cot θ = Side adjacent to the angle/ side opposite to the angle
The trigonometric ratios are used to find missing sides of right triangles when at least one side length and one angle measure are given. Whereas the inverse trigonometric ratios are used to find the missing angle in a right triangle when two sides are given.
Browse the 12th class video on Formulas Of Inverse Trigonometric Functions
Trigonometric ratios and inverse trigonometric ratios are used in oceanography for calculating the height of tides in oceans. The sine and cosine functions are fundamental to the theory of periodic functions, those that describe sound and light waves. Trigonometry is also an important part of calculus. These ratios can be used to measure the height of a building or mountains. These ratios are also used in video games and construction. Flight engineers have to take into account their speed, distance, and direction along with the speed and direction of the wind. The wind plays an important role in how and when a plane will arrive where ever needed this is solved using vectors to create a triangle using trigonometry to solve.
Trigonometry is used to find the components of vectors, model the mechanics of waves (both physical and electromagnetic) and oscillations, sum the strength of fields and use dot and cross products. In criminology, trigonometry can help to calculate a projectile’s trajectory. Marine biologists can use these ratios to find out how light levels at different depths affect the ability of algae to photosynthesize. These ratios are also used in navigation in order to pinpoint a location. Also, trigonometry has its applications in satellite systems.
A Trigonometric Function is invertible if it is one to one and onto. These trigonometric ratios are not one to one and onto over their natural domains and ranges and hence, their inverses do not exist. For example:
For sine function, the domain is the set of all real numbers and range is . We have
So, the sine function is not one to one, and hence, it’s inverse does not exist.
If the domain of sine function is restricted tothen it becomes one to one and onto and hence, invertible. Sine function restricted to any of the intervals:
etc., is one to one and onto with range as.
Therefore, the inverse of sine function exists with the domain as and range as any of the intervals:
etc. The branch with range
is known as the principal value branch.
So, domains and ranges of all the trigonometric ratios are restricted to ensure the existence of their inverses.
RANGE AND DOMAIN OF TRIGONOMETRIC FUNCTIONS
The domain and ranges of inverse trigonometric functions are given in the following table:
sin-1 x, cos-1 x, tan-1 x, cosec-1 x, sec-1 x, and cot-1 x should not be confused
with (sin x)-1 , (cos x)-1 , (tan x)-1 , (cosec x)-1 , (sec x)-1 , and (cot x)-1 , respectively.
IMPORTANT FORMULAS OF INVERSE TRIGONOMETRIC FUNCTIONS – CBSE Class 12 Maths
Inverse trigonometric functions formulas
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