# Trig function of arc trig functions and the reverse

• Philosophaie
In summary: Similarly, cos(arcsin(x)) is the adjacent side (1) divided by the hypotenuse \sqrt{1- x^2}. You should be able to do the rest yourself.In summary, the formulas for cos(arcsin(x)) and tan(arccos(x)) can be derived using the Pythagorean theorem and the definitions of sine, cosine, and tangent. However, the formulas for tan(arcsin(x)), sin(arctan(x)), cos(arctan(x)), arcsin(cos(x)), arcsin(tan(x)), arccos(sin(x)), arccos(tan(x)), arctan(sin(x)), and arctan(cos(x)) may not always have solutions depending on the
Philosophaie
I know the sin(arccos(x)) = (1-x^2)^0.5

I was wondering what some of the others are:

cos(arcsin(X))
tan(arcsin(X))
tan(arccos(x))
sin(arctan(x))
cos(arctan(x))

also the reverse:

arcsin(cos(x))
arcsin(tan(X))
arccos(Sin(X))
arccos(tan(X))
arctan(sin(X))
arctan(cos(X))

Let me do two:

$$\cos(arcsin(x))=\sqrt{1-\sin(arcsin(x))}=\sqrt{1-x^2}$$

and

$$\tan(arccos(x))=\sqrt{\frac{1}{\cos^2(arccos(x))}-1}=\sqrt{\frac{1}{x^2}-1}$$

I'll let you find out the other ones...

are these always possible? I mean take

arccos(tan(x))

for example, the cosine of an angle is always between 0 and 1, and so, the argument to the arccos function should be a number between 0 and 1...but the tangent of an angle can get pretty large...so, I think these is no solution here...same for others.

micromass said:
Let me do two:

$$\tan(arccos(x))=\sqrt{\frac{1}{\cos^2(arccos(x))}-1}=\sqrt{\frac{1}{x^2}-1}$$

Not if arccos(x) is in the second quadrant.

As long as your "angles" are in the first quadrant (so you don't have multi-value problems), you can get all of those formulas by constructing an appropriate right triangle.

For example, to get sin(arctan(x)), imagine a right triangle with "opposite side" x and "near side" 1 (so that the tangent of the angle opposite side "x" is x/1= x and the angle is arctan(x)). By the Pythagorean theorem, it will have "hypotenuse" $\sqrt{x^2+ 1}$. Sine is "opposite side over hypotenuse" so $sin(arctan(x))= \frac{x}{\sqrt{x^2+ 1}}$.

## What are trigonometric functions and how are they related to arc trigonometric functions?

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the lengths of its sides. They include sine, cosine, tangent, cotangent, secant, and cosecant. Arc trigonometric functions are the inverse of these functions and relate the angles to the ratios of the sides.

## What is the difference between a trigonometric function and an arc trigonometric function?

A trigonometric function takes an angle as input and gives a ratio of sides as output, while an arc trigonometric function takes a ratio of sides as input and gives an angle as output.

## What are the common trigonometric functions and their corresponding arc trigonometric functions?

The common trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Their corresponding arc trigonometric functions are arcsine (arcsin), arccosine (arccos), arctangent (arctan), arccotangent (arccot), arcsecant (arcsec), and arccosecant (arccsc).

## How are trigonometric functions and arc trigonometric functions used in real-world applications?

Trigonometric functions are used to solve problems involving right triangles, such as finding the height of a building or the distance between two points. Arc trigonometric functions are used to find the angle measurements in a triangle when the lengths of the sides are known. They are also used in fields like physics, engineering, and astronomy to model and analyze various phenomena.

## What are the identities and properties of trigonometric and arc trigonometric functions?

Trigonometric identities are equations that are true for all values of the variables involved. They include properties such as the Pythagorean identity and the sum and difference identities. Arc trigonometric functions also have identities and properties, such as the inverse property and the double angle identity. These identities and properties are important in solving equations and simplifying expressions involving trigonometric and arc trigonometric functions.

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