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In summary, the conversation discusses ways to simplify expressions involving inverse trigonometric functions, specifically cos(sin-1(x)). The hint given suggests using the relation ArcSin((1-x^2)^0.5)=ArcCos(x) to solve for the expression in terms of a right triangle and the angle \theta.
  • #1
Astudious
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Is there any way I can reduce or simplify expressions like cos(sin-1(x)), sin(cos-1(x)), cos(tan-1(x)), tan(cot-1(x)) etc.? (I refer to the arc functions, i.e. inverses, by the superscript -1, not the reciprocals)
 
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  • #2
Hint: sin(sin-1(x)) = x and cos(w) = sin(w+π/2).
 
  • #3
You can use this relation:
ArcSin((1-x^2)^0.5)=ArcCos(x)
 
  • #4
Astudious said:
Is there any way I can reduce or simplify expressions like cos(sin-1(x))

Draw a right triangle with an angle [itex]\theta[/itex], opposite side length of x and hypotenuse length 1. Now from this triangle, by definition,

[tex]\sin\theta = x[/tex]
hence
[tex]\theta = \sin^{-1}x[/tex]

So then what is [itex]\cos\left(\sin^{-1}x\right)[/itex], or more simply, [itex]\cos\theta[/itex] ?
 
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What are trigonometric equations?

Trigonometric equations are mathematical expressions that involve trigonometric functions, such as sine, cosine, and tangent. They are used to solve problems related to triangles and other geometric shapes.

What is the general form of a trigonometric equation?

The general form of a trigonometric equation is f(x) = a sin(bx + c) + d, where a, b, c, and d are constants and f(x) represents the trigonometric function.

How do you solve a trigonometric equation?

To solve a trigonometric equation, you need to isolate the trigonometric function on one side of the equation and use algebraic techniques to solve for the variable. You may also need to use trigonometric identities and special values to simplify the equation.

What are the common types of trigonometric equations?

Some common types of trigonometric equations include linear equations, quadratic equations, and equations involving multiple trigonometric functions. These equations may also involve inverse trigonometric functions, such as arcsine, arccosine, and arctangent.

Why are trigonometric equations important?

Trigonometric equations are important in many fields of science and engineering, such as physics, astronomy, and geometry. They are used to solve real-world problems involving angles, distances, and other geometric properties.

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