A function in mathematics is a relation between a set of inputs and a set of possible outputs. Each input is related to exactly one output. Functions are often represented by equations or graphs and they are used to model real-world situations in various fields like physics, economics and engineering.
Class 12 Maths Chapter 2 Inverse Trigonometric Functions covers the inverse of trigonometric functions like sine and cosine and tangent or cosecant and secant and cotangent. It helps in finding angles when the value of a trigonometric function is given. The chapter includes domains ranges graphs and solving equations involving inverse trigonometric functions.
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We are given to determine sin(tan⁻¹x), where |x| < 1 .
Step 1: Let us take θ = tan⁻¹x .
This gives us the following equations:
tan(θ) = x
Since tangent of any angle is the ratio of the opposite side to the adjacent side, we can depict it in a right triangle as below:
-Opposite side= x
Adjacent side = 1
Step2: Applying Pythagorean theorem
To find the hypotenuse, use the Pythagorean theorem:
Hypotenuse = √(1² + x²) = √(1 + x²)
Step 3: Calculate sin(θ)
We know that:
sin(θ) = Opposite / Hypotenuse = x / √(1 + x²)
Final Answer:
Thus, the value of sin(tan⁻¹x) is:
x / √(1 + x²)
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