Solve the equation root 3 sec theta + 2 is equal to zero and cosec theta is equal to two

To tackle the equations you've provided, we need to break them down and solve them step by step. Let's look at each part carefully, starting with the first equation:

Solving the First Equation

The first equation is:

√3 sec(θ) + 2 = 0

To isolate sec(θ), we can rearrange the equation:

• Subtract 2 from both sides:
• √3 sec(θ) = -2

Next, we divide both sides by √3:

sec(θ) = -2/√3

Recall that sec(θ) is the reciprocal of cos(θ), which means:

cos(θ) = -√3/2

Now, let's determine the angles θ that satisfy this cosine value. The cosine function is negative in the second and third quadrants:

• In the second quadrant: θ = 150° or 5π/6 radians.
• In the third quadrant: θ = 210° or 7π/6 radians.

Summary of First Equation Solutions

The solutions for the first equation are:

• θ = 150° + 360°k or θ = 210° + 360°k, where k is any integer.

Solving the Second Equation

Now, let’s move on to the second equation:

cosec(θ) = 2

Cosecant is the reciprocal of sine, so we can rewrite this as:

sin(θ) = 1/2

To find the angles θ that yield this sine value, we consider where sine is positive, which occurs in the first and second quadrants:

• In the first quadrant: θ = 30° or π/6 radians.
• In the second quadrant: θ = 150° or 5π/6 radians.

Summary of Second Equation Solutions

The solutions for the second equation are:

• θ = 30° + 360°k or θ = 150° + 360°k, where k is any integer.

Combining the Solutions

Now, let’s analyze the solutions from both equations:

• From the first equation, we found θ = 150° and θ = 210°.
• From the second equation, we found θ = 30° and θ = 150°.

The common solution between the two equations is:

θ = 150° + 360°k

In conclusion, the values of θ that satisfy both equations are those of the form 150° + 360°k, where k is any integer. This means that the angle can be expressed in a periodic manner, providing multiple solutions over the range of angles. If you have any further questions or need clarification on any steps, feel free to ask!