solve equation in interval.
sec x = -3
-¦Ð ¡Ü x < ¦Ð
the answer is x = -1.911 and 1.911
could someone please write the steps to solving this
interval is :
pi is less than or equal to (x) is less than pi
To solve the equation sec(x) = -3 in the given interval -π/2 ≤ x ≤ π/2, you can follow these steps:
Step 1: Recall that sec(x) is the reciprocal of the cosine function, so we have sec(x) = 1/cos(x).
Step 2: Replace sec(x) with 1/cos(x), so we now have 1/cos(x) = -3.
Step 3: Multiply both sides of the equation by cos(x) to get rid of the denominator, giving us 1 = -3cos(x).
Step 4: Divide both sides of the equation by -3 to isolate the cosine function, resulting in cos(x) = -1/3.
Step 5: Find the inverse cosine (also known as arccosine) of both sides to solve for x. In this case, since -π/2 ≤ x ≤ π/2, we can use the principal value of the inverse cosine.
Step 6: Use a calculator or a table to find the principal value of the inverse cosine of -1/3, which is approximately 1.911 radians.
Step 7: Since the cosine function is symmetric about the y-axis (cos(-x) = cos(x)), we can identify that -1.911 is the negative value of the solution in the given interval.
Step 8: Therefore, the two solutions in the given interval are x = -1.911 and x = 1.911.
Note: In other intervals, where the range of x is different, you may need to adjust the solutions accordingly.