(a) Solve the equation
sin x= cos 140 for -180 < x< 180
To solve the equation sin(x) = cos(140), we first need to find the value of x that satisfies this equation within the given range of -180 < x < 180.
To do this, we can use the fact that sin(x) = cos(90 - x). Therefore, sin(x) = cos(140) can be rewritten as sin(x) = cos(90 - 140) = cos(-50).
Now, we can compare the values of sine and cosine for -180 < x < 180 to find the value of x that satisfies sin(x) = cos(-50).
In this range, sin(x) is positive for x in the first and second quadrants, while cos(x) is positive for x in the first and fourth quadrants.
Since we need sin(x) = cos(-50), which is positive, we can conclude that the solution lies in the first quadrant.
Now, we can use inverse trigonometric functions to find the angle x that satisfies cos(-50).
cos^(-1)(cos(-50)) = -50
Therefore, x = -50 is the solution to the equation sin(x) = cos(140) within the given range of -180 < x < 180.
To solve the equation sin x = cos 140, we can use the trigonometric identity sin(x) = cos(90° - x).
Therefore, sin x = cos(90° - x)
Comparing this with the given equation sin x = cos 140, we can equate the angles:
x = 90° - 140°
x = -50°
So, the solution to the equation sin x = cos 140, for -180° < x < 180°, is x = -50°.
To solve the equation sin x = cos 140, you need to find the value of x that satisfies this equation within the given range of -180 < x < 180.
Step 1: Rewrite the equation using the identity sin(90 - θ) = cos θ.
sin x = cos 140 can be rewritten as sin x = sin (90 - 140).
Step 2: Simplify the equation.
sin x = sin (-50)
Step 3: Apply the inverse sine function (sin⁻¹) to both sides of the equation to isolate x.
x = sin⁻¹(sin (-50))
Step 4: Evaluate the expression on the right-hand side.
x = sin⁻¹(sin (-50))
Since the given range is -180 < x < 180, the principal value of x lies within this range.
To find the x-value, you can use a scientific calculator or an online calculator that has the inverse sine function.
Using a calculator, find the inverse sine of sin (-50):
sin⁻¹(sin (-50)) = -50°
Therefore, the solution to the equation sin x = cos 140 in the range -180 < x < 180 is x = -50°.