Why does sin(arctan(x)) simplify to x / sqrt(1 + x²)?

If θ = arctan(x), then tan(θ) = x. In a right triangle, if the opposite side is x and the adjacent side is 1, the hypotenuse is √(1+x²). Therefore, sin(θ) = opposite/hypotenuse = x/√(1+x²).

How to visualize composite trig functions?

Start with the inner function (e.g., arctan(x)) to get an angle, then apply the outer function (e.g., sin) to that angle. You can also use right triangle relationships to simplify these expressions.

🔢 sin(arctan(x)) Calculator

Compute expressions like sin(arctan(x)), cos(arctan(x)), and tan(arcsin(x)).

sin(arctan(x)) Calculator

Value(Any real number)

Composite trigonometric functions combine multiple trigonometric operations. For example, sin(arctan(x)) simplifies to x/√(1+x²). Our calculator handles these composite functions and shows both the simplified form and the numerical result.

Quick Examples

Common composite trigonometric expressions.

sin(arctan(1)) = √2/2 ≈ 0.707

Since arctan(1) = 45°, sin(45°) = √2/2.

sin(arctan(0)) = 0

Since arctan(0) = 0°, sin(0°) = 0.

cos(arctan(1)) = √2/2 ≈ 0.707

Since arctan(1) = 45°, cos(45°) = √2/2.

Use Cases

Common applications of composite trigonometric functions.

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High School Trigonometry

Simplifying complex expressions
Understanding function composition
Solving trigonometric equations

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Calculus

Trigonometric substitution
Integration techniques
Derivative calculations

Frequently Asked Questions

Common questions about composite trigonometric functions.

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Last updated: 2025-10-21 — Compiled and reviewed by Calvin (Math Research, FreeCalculators.app)