The question is:
tan(arcsinx)=
I know that inverse of sin is 1/sqrt(1-x^2) however this question still confuses me. The outcome is x/sqrt(1-x^2), how did the tan create the additional x ?
oops double posted
make a sketch of a right-angled triangle, where the base angle is Ø
consider arcsin x
that means, in our triangle state the angle Ø so that sin Ø = x or sin Ø = x/1
so in your triangle label the adjacent side x and the hypotenuse 1
by Pythagoras, y^2 + x^2 = 1^2
y^2 = 1 - x^2
y = √(1 - x^2) , you had that
now tan(arcsin (x) )
= tan Ø
= y/x
= √(1-x^2)/x
To solve for `tan(arcsinx)`, we first need to understand the relationship between the trigonometric functions `sin` and `tan`.
In a right triangle, the `sin` function represents the ratio of the length of the side opposite the angle to the length of the hypotenuse. On the other hand, the `tan` function represents the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Now, let's find `tan(arcsinx)` step by step.
Step 1: Let `y = arcsinx`
This means that `sin(y) = x`, where `x` is some value between -1 and 1.
Step 2: Draw a right triangle.
We can imagine a right triangle with one of the angles `y` and `x` as the length of the side opposite angle `y`.
Step 3: Find the measurements of the sides of the right triangle.
Since `sin(y) = x`, the length of the side opposite angle `y` is `x`, and the hypotenuse of the triangle is 1. Based on the Pythagorean theorem, we can find the length of the side adjacent to angle `y`.
Using the Pythagorean theorem, we have:
(x)^2 + (side adjacent to angle y)^2 = (hypotenuse)^2
x^2 + (side adjacent to angle y)^2 = 1^2
(side adjacent to angle y)^2 = 1 - x^2
(side adjacent to angle y) = √(1 - x^2)
Step 4: Calculate `tan(arcsinx)`.
Now that we have the side lengths of the triangle, we can find `tan(arcsinx)` using the definition of the `tan` function:
`tan(arcsinx)` = (side opposite angle y) / (side adjacent to angle y)
= x / √(1 - x^2)
So, the outcome of `tan(arcsinx)` is indeed `x / √(1 - x^2)`, which includes the additional `x` as calculated using the relationship between the `sin` and `tan` functions within a right triangle.