Hi,
Find the exact value of the following expression:
Sin^-1(1/radical2)
(Its inverse sin(1/radical2)
Direction -(Type your answers in interval notation. Type an exact answer using pi as need)
Please show how you got the answers so I can use this as a reference for others. THANK YOU.
There are no intervals involved.
since sin(π/4) = 1/√2
the principal value of arcsin,
Sin^-1(1/√2) = π/4
But, since there are many other angles,
sin^-1(1/√2) = π/4 or 3π/4
and since the period of sin(x) is 2π,
sin^-1(1/√2) = π/4 + 2nπ or 3π/4 + 2nπ
Or, since those values are in QI and QII, symmetric with the y-axis,
sin^-1(1/√2) = (4n+1)(π/2) ± π/4
To find the exact value of the expression sin^(-1)(1/√2), let's understand the concept of inverse trigonometric functions.
The inverse sine function (sin^(-1) or arcsin) returns an angle whose sine is equal to the given value. In other words, if sin(x) = y, then sin^(-1)(y) = x.
In this case, sin^(-1)(1/√2) means we need to find an angle whose sine is equal to 1/√2.
To find this angle, we can use the trigonometric ratios related to the special angles of a 45-45-90 right triangle.
In a 45-45-90 triangle, two sides are equal, and the ratio of the length of the legs to the length of the hypotenuse is √2.
Let's assume the angle we're looking for is θ. We can create a right triangle with one leg equal to 1 and the hypotenuse equal to √2.
Using the Pythagorean theorem, we find the other leg: (1^2) + (other leg^2) = (√2^2)
Simplifying this equation, we get:
1 + (other leg^2) = 2
(other leg^2) = 2 - 1
(other leg^2) = 1
other leg = 1
So, we have a right triangle with legs of length 1 and a hypotenuse of length √2. By definition, the sine of θ is equal to the opposite side divided by the hypotenuse:
sin θ = opposite/hypotenuse
sin θ = 1/√2
Therefore, we have found that sin θ = 1/√2.
Now, to find the exact value of θ, we need to find the angle whose sine is equal to 1/√2.
Using a unit circle, we can identify the angle that satisfies this condition. On the unit circle, the point (cos θ, sin θ) represents the angle θ. Since sin θ is positive and equal to 1/√2, we know that the angle θ lies in the first and second quadrants.
To determine the value of θ, we can use the reference angle, which is the acute angle between the terminal side of the angle θ and the x-axis.
The reference angle for sin^(-1)(1/√2) is π/4 (45 degrees). This is because the sine function is positive in the first and second quadrants and reaches its highest positive value of 1/√2 at π/4.
Since θ belongs to the first and second quadrants, we can find its exact value by applying the following formula:
θ = π/4 + 2kπ or θ = π - π/4 + 2kπ, where k is an integer.
Thus, the exact value of sin^(-1)(1/√2) is π/4 + 2kπ and π - π/4 + 2kπ, where k is an integer.
In interval notation, the solution can be written as:
[π/4 + 2kπ, π - π/4 + 2kπ], where k is an integer.