Write an algebraic expression that is equivalent to the expression (Sketch right triangle)
sec[arcsin(x-1)]
An angle whose sine is x-1 has a cosine of sqrt[1 -(x-1)^2]' = sqrt(-x^2+2x)
Its secant is 1/sqrt[x(2-x)]
x is limited to numbers between 0 and 2.
To find an equivalent algebraic expression for sec[arcsin(x-1)], we can start by considering the definitions of the trigonometric functions involved.
The arcsin (or sin^(-1)) function gives us an angle whose sine is equal to its argument. In this case, arcsin(x-1) gives us an angle whose sine is equal to (x-1). Let's call this angle theta.
The secant function (sec) is the reciprocal of the cosine function (cos). So, to find an algebraic expression for sec[arcsin(x-1)], we need to find the cosine of theta.
To find cosine, we can use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1. Since we know the value of sin(theta) as (x-1), we can substitute it into the equation and solve for cos(theta).
sin^2(theta) + cos^2(theta) = 1
(x-1)^2 + cos^2(theta) = 1
cos^2(theta) = 1 - (x-1)^2
cos^2(theta) = 1 - (x^2 - 2x + 1)
cos^2(theta) = 2x - x^2
Now that we have an expression for cos^2(theta), we can find the reciprocal to get sec(theta).
sec(theta) = 1/cos(theta) = 1 / sqrt(2x - x^2)
Therefore, an algebraic expression equivalent to sec[arcsin(x-1)] is 1 / sqrt(2x - x^2).
To simplify the expression sec[arcsin(x-1)], let's start by understanding the definitions of sec and arcsin.
1. arcsin: The arcsine function (arcsin) returns the angle whose sine is a given number. It's the inverse of the sine function. arcsin(x) is also written as sin^(-1)(x).
2. sec: The secant function (sec) is the reciprocal of the cosine function. It can be defined as sec(x) = 1/cos(x).
Now, let's work on simplifying the expression step by step.
Step 1: Replace arcsin(x-1) with a variable.
Let's say we replace arcsin(x-1) with a variable, let's say 'y'. So the expression now becomes sec(y).
Step 2: Determine the value of y.
Since we replaced arcsin(x-1) with y, we need to find the value of y that makes it equal to (x-1). This means we need to find the sine of y, which is (x - 1).
Step 3: Find sin(y) using the given information.
We know that sin^(-1) represents the inverse of the sine function. So y = sin^(-1)(x - 1).
Step 4: Express the right triangle's relationship between sides in terms of x.
To express the relationship between sides in terms of x, let's draw a right triangle.
Let's assume one angle of the right triangle is y, and the opposite side to that angle has a length of (x - 1). We can label the opposite side as 'opposite,' the adjacent side as 'adjacent,' and the hypotenuse as 'hypotenuse.'
Using the Pythagorean theorem, we can find the hypotenuse.
hypotenuse = √(opposite^2 + adjacent^2)
Here, the hypotenuse is equal to 1 because secant is the reciprocal of cosine, and cosine is adjacent/hypotenuse. And since sec(y) = 1/cos(y), sec(y) = 1/hypotenuse.
So, sec(y) = 1/1, which simplifies to sec(y) = 1.
Hence, the simplified expression is 1. Therefore, an algebraic expression equivalent to sec[arcsin(x-1)] is 1.