1. Find the exact value
3sin(invertangent (-1))
I got 0.
Given 0<x<orequalto 1, determine the value of inversine (x) + inversetan (squareroot(1-x^2)/x)
I really have no idea how to do that. I am trying to draw a diagram but cant figure it out.
Thanks!
To find the exact value of 3sin(arctan(-1)), we can use the trigonometric identities and properties to simplify the expression.
The arctan function gives us the angle whose tangent is equal to a given value. In this case, arctan(-1) represents the angle whose tangent is -1. Since tangent is negative in the third and fourth quadrants, we can consider an angle of -45 degrees or -π/4 radians.
Now, we can calculate the value of sin(-π/4). Remember that the sine function gives us the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.
Using the unit circle, we can determine that sin(-π/4) is -√2/2.
So, the exact value of 3sin(arctan(-1)) is 3 * (-√2/2), which simplifies to -3√2/2 or approximately -2.121.
Now let's move on to the second question.
To find the value of inversetan (sqrt(1-x^2)/x) + inversesin(x) where 0 < x ≤ 1, we need to use the properties and definitions of inverse trigonometric functions.
First, let's focus on inversetan(sqrt(1-x^2)/x). We have the square root of (1-x^2) in the numerator and x in the denominator.
Using the unit circle or the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can find that (1-x^2) is equal to sin^2(theta), where theta represents an angle.
So inversetan(sqrt(1-x^2)/x) can be rewritten as inversetan(sin(theta)/x). Since x is positive, we can assume theta is positive as well.
Now, let's consider inversetan(sin(theta)/x) + inversesin(x).
Using the definitions of inverse trigonometric functions, inversetan(sin(theta)/x) represents the angle whose tangent is sin(theta)/x, and inversesin(x) represents the angle whose sine is x.
We can use the trigonometric identity tan(theta) = sin(theta)/cos(theta) to rewrite inversetan(sin(theta)/x) as inversetan(tan(theta)).
The inverse of the tangent and tangent functions cancel each other out, so we are left with theta.
Therefore, inversetan(sin(theta)/x) + inversesin(x) simplifies to just theta + theta, which is equal to 2theta.
Since we have x as an upper limit in the question (0 < x ≤ 1), we need to find the corresponding values of theta in the range 0 ≤ theta ≤ π/2.
We know that x = sin(theta) and 0 < x ≤ 1, so the values of x correspond to the values of theta in the first quadrant.
To find the value of theta, we can use the inverse sine function, which gives us the angle whose sine is equal to a given value.
Therefore, theta = inversesin(x) = inversesin(sin(theta)) = theta.
So, we can say that 2theta is equal to 2 * theta.
Therefore, the value of inversetan(sqrt(1-x^2)/x) + inversesin(x) where 0 < x ≤ 1 is simply 2 * theta, which is 2 times the angle corresponding to x.
I hope this clears up the confusion and helps you understand how to solve these types of trigonometric problems.