solve and round your answer to 4 dec.
sinx= .4 in [0,2pi]
secx= -3 in [0,pi]
To solve the equation sin(x) = 0.4 in the interval [0, 2π], we need to find the values of x that satisfy this equation.
Step 1: Determine the reference angle:
Since the sine function is positive in both the first and second quadrants, we can look for the reference angle in the first quadrant.
arcsin(0.4) ≈ 0.4115
Step 2: Find the principal solution:
The principal solution can be found using the reference angle in the first quadrant:
x = 0.4115
Step 3: Find the general solutions:
The general solutions can be found by adding multiples of the period (2π) to the principal solution:
x = 0.4115 + 2πn, where n is an integer.
Therefore, the general solutions for sin(x) = 0.4 in the interval [0, 2π] are:
x ≈ 0.4115, 6.853 + 2πn, where n is an integer.
To solve the equation sec(x) = -3 in the interval [0, π], we similarly need to find the values of x that satisfy this equation.
Step 1: Determine the reference angle:
Since the secant function is negative in the interval [0, π], we need to look for the reference angle in the second quadrant.
arcsec(-3) ≈ 2.0944
Step 2: Find the principal solution:
The principal solution can be found using the reference angle in the second quadrant:
x = π - 2.0944 ≈ 1.0472
Step 3: Find the general solutions:
The general solutions can be found by adding multiples of the period (2π) to the principal solution:
x = 1.0472 + 2πn, where n is an integer.
Therefore, the general solutions for sec(x) = -3 in the interval [0, π] are:
x ≈ 1.0472 + 2πn, where n is an integer.
Please note that the answers are rounded to 4 decimal places as requested.