Solve each equation for theta, giving a general formula for all of the solutions:

a. tan(theta)=-1
b. sin(theta/2)=1/2

tan(-π/4) = -1

so, θ = -π/4 + nπ

sin(π/6) = 1/2, so θ/2 = π/6,5π/6,...
θ = π/3, 5π/3, ... = π/3 + 4nπ, 5π/3 + 4nπ

a. To solve the equation tan(theta) = -1, we will use the inverse tangent function (also known as arctan or atan). The inverse tangent function allows us to find the angle that has a given tangent value.

1. Start by applying the inverse tangent function to both sides of the equation:
arctan(tan(theta)) = arctan(-1)

2. The inverse tangent function and the tangent function are inverse operations, so they cancel each other out:
theta = arctan(-1)

3. Calculate the arctan(-1). This can be done using a calculator, and it will give you the value of the reference angle. The reference angle is the principal value between -π/2 and π/2, where the tangent function is defined.

4. When arctan(-1) is evaluated, it gives you -π/4 (or -45 degrees) as the reference angle.

5. However, we need to take into account that the tangent function has a period of π. This means that there are infinitely many solutions to the equation tan(theta) = -1.

6. To find the general formula for all solutions, you can add integer multiples of π to the reference angle (-π/4). Therefore, the general formula for theta is:
theta = -π/4 + nπ, where n is any integer.

b. To solve the equation sin(theta/2) = 1/2, we will use the inverse sine function (also known as arcsin or asin). The inverse sine function allows us to find the angle that has a given sine value.

1. Start by applying the inverse sine function to both sides of the equation:
arcsin(sin(theta/2)) = arcsin(1/2)

2. The inverse sine function and the sine function are inverse operations, so they cancel each other out:
theta/2 = arcsin(1/2)

3. Calculate the arcsin(1/2). This can be done using a calculator, and it will give you the value of the reference angle. The reference angle is the principal value between -π/2 and π/2, where the sine function is defined.

4. When arcsin(1/2) is evaluated, it gives you π/6 (or 30 degrees) as the reference angle.

5. However, we need to take into account that the sine function has a period of 2π. This means that there are infinitely many solutions to the equation sin(theta/2) = 1/2.

6. To find the general formula for all solutions, you can add integer multiples of 2π to the reference angle (π/6). Therefore, the general formula for theta is:
theta/2 = π/6 + 2nπ, where n is any integer.

7. To obtain the full set of solutions for theta, multiply both sides of the equation by 2:
theta = 2π/6 + 2nπ

Simplifying the expression, we get:
theta = π/3 + nπ, where n is any integer.