Question

Trignometric equation for all values sec4x+2=0

Answer 1

To solve the equation sec4x+2=0, isolate the secant function by subtracting 2 from both sides. Then, rewrite the equation as cos4x = -1/2 and find the reference angle. Finally, use the properties of cosine to find the general solution for x.

Step-by-step explanation:

The given equation is sec4x+2=0. To solve this equation, we need to use properties of trigonometric functions. Here are the steps to find the solutions:

1. Start by isolating the secant function by subtracting 2 from both sides: sec4x = -2.
2. Since secant is the reciprocal of cosine, we can rewrite the equation as cos4x = -1/2.
3. Next, find the reference angle by taking the inverse cosine of -1/2: 4x = 2.094 (in radians) or 120°.
4. Now, we need to find the general solution for x. Since cosine has a period of 2π, we can add 2πn to the reference angle, where n is an integer. This gives us the general solution: x = 0.523 + nπ/2 (in radians) or x = 30 + n*90°.

Therefore, the equation sec4x+2=0 has infinitely many solutions given by x = 0.523 + nπ/2 (in radians) or x = 30 + n*90°, where n is an integer.