(a) Solve the equation
sin x= cos 140 for -1809 < x< 180
To solve the equation sin x = cos 140, we need to find the values of x that satisfy this equation.
Since sin x = cos (90 - x), we can rewrite the equation as sin x = sin (90 - 140) = sin (-50).
Using the property of sine function, we know that sin (-x) = -sin x. Therefore, sin (-50) = -sin 50.
So, the equation sin x = sin (-50) simplifies to sin x = -sin 50.
Now, we need to find the values of x that satisfy this equation. We can do this by finding the reference angle for both sin x and -sin 50, and then adding/subtracting the reference angle from multiples of 360 degrees to get the solutions.
The reference angle for sin x is 50 degrees, and the reference angle for -sin 50 is also 50 degrees.
Adding/subtracting multiples of 360 degrees, we have the solutions:
x = 50 + 360n, where n is an integer
x = -50 + 360n, where n is an integer
However, we are only interested in the values of x within the given domain of -1809 < x < 180.
Checking the solutions within this domain, we find that:
x = 50 + 360n | -1809 < x < 180
-1759 < 50 + 360n < 180
This inequality holds for n = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.
Therefore, the solutions for the equation sin x = cos 140 within the given domain are:
x = 50 + 360n, where n = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
x = -50 + 360n, where n = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
To solve the equation sin(x) = cos(140), we can use the identity sin(x) = cos(90 - x).
cos(140) = sin(90 - x)
Now, let's solve for x:
140 = 90 - x
x = 90 - 140
x = -50
Therefore, the solution to the equation sin(x) = cos(140) is x = -50.
To solve the equation sin(x) = cos(140), we need to find the values of x that satisfy this equation within the given range of -180 to 180.
Let's start by writing sin(x) and cos(140) in terms of their trigonometric identities:
sin(x) = cos(90 - x)
cos(140) = sin(90 - 140)
Now, we can rewrite the equation as:
cos(90 - x) = sin(90 - 140)
Using the trigonometric identity: cos(a) = sin(90 - a), we can simplify the equation to:
sin(90 - x) = sin(90 - 140)
Now, we can equate the angles inside the sine functions:
90 - x = 90 - 140
Simplifying further:
-x = -140
Dividing both sides by -1, we have:
x = 140
So, the solution to the equation sin(x) = cos(140) within the given range of -180 to 180 is x = 140.