I'm having a lot of trouble on this word problem. Can someone help me plz?
To define the inverse sine function, we restrict the domain of sine to the interval ______. On this interval the sine function is one-to-one, and its inverse function
sin^−1 is defined by sin^−1 x = y ⇔ sin_______=_______.
For example,
sin−1 (1/2)=_______ because sin_____=______.
(b) To define the inverse cosine function we restrict the domain of cosine to the interval_______. On this interval the cosine function is one-to-one and its inverse function
cos^−1 is defined by
cos−1 x = y ⇔ cos_______=________.
For example,
cos^−1(1/2)=_______ because cos_____=______.
.
I don't agree the sine and inverse sine functions are one to one. The sine function is cyclic.
arcsin(1/2)=30 deg and 150 and ...
on the interval [-90,90] sin^-1 is 1:1
For cos^-1, the domain is [0,180]
But you probably want radians instead.
It showed up as wrong on my sheet
can you see how unhelpful that response is? What did you enter, and why was it wrong?
So basically, how would you restrict the sin/cos graph so that it becomes 1 to 1? you section off the graph via interval notations. So it would like half of a period but with its head and end cut off. (at the highest point and lowest point, so it passes the HLT - Horizontal Line Test)
... restrict the domain of cosine to the interval [-pi/2, pi/2].
you can figure the rest out.
To answer these questions, we need to understand the concepts of inverse sine and inverse cosine functions and their domains. Let me explain how to find the answers.
(a) To define the inverse sine function, we restrict the domain of sine to the interval [-π/2, π/2]. On this interval, the sine function is one-to-one, meaning it has a unique output for every input. The inverse sine function is denoted as sin^−1(x), also known as arcsin(x), and is defined by sin^−1(x) = y if and only if sin(y) = x.
For example, to find sin^−1(1/2), we need to find the angle y such that sin(y) = 1/2. We know that sin(π/6) = 1/2, so sin^−1(1/2) = π/6.
(b) To define the inverse cosine function, we restrict the domain of cosine to the interval [0, π]. On this interval, the cosine function is one-to-one. The inverse cosine function is denoted as cos^−1(x), also known as arccos(x), and is defined by cos^−1(x) = y if and only if cos(y) = x.
For example, to find cos^−1(1/2), we need to find the angle y such that cos(y) = 1/2. We know that cos(π/3) = 1/2, so cos^−1(1/2) = π/3.
In summary:
(a) The inverse sine function sin^−1(x) is defined by sin^−1(x) = y if and only if sin(y) = x. For example, sin^−1(1/2) = π/6 because sin(π/6) = 1/2.
(b) The inverse cosine function cos^−1(x) is defined by cos^−1(x) = y if and only if cos(y) = x. For example, cos^−1(1/2) = π/3 because cos(π/3) = 1/2.
I hope this explanation helps you understand how to find the answers to these word problems! If you have any further questions, feel free to ask.