Find the value of theta:

-810Cos(theta) + 135 + 2430 Sin(theta)Cos(theta) = 0

Can someone please show me how to simplify it to find the value of theta? Thanks!

it is easy to see That theta is PI/2 is one solution, and 3PI/2 is a second solution. Whenever CosTheta is zero...

There may be a second solution, it is not "easy" to find it except by graphing.
graph y=-810Cos(theta) + 135 + 2430 Sin(theta)Cos(theta)
and the solutions if any will be when y is zero

To simplify the equation and find the value of theta, we can follow these steps:

Step 1: Rearrange the equation
-810Cos(theta) + 135 + 2430Sin(theta)Cos(theta) = 0

Step 2: Combine like terms
To simplify, we need to combine the terms involving cosine(theta) and sine(theta) together.

-810Cos(theta) + 2430Sin(theta)Cos(theta) = -135 (subtract 135 from both sides)

Step 3: Factor out the common term
In the equation, we notice that Cos(theta) is common to both terms. We can factor it out to simplify further.
Cos(theta) (-810 + 2430Sin(theta)) = -135

Step 4: Divide both sides by the remaining term
Divide both sides by (-810 + 2430Sin(theta)) to isolate the Cos(theta) term.

Cos(theta) = -135 / (-810 + 2430Sin(theta))

Step 5: Solve for Cos(theta)
By now, we have simplified the equation to isolate Cos(theta). Next, solve for Cos(theta) by finding the reciprocal of both sides.

1 / Cos(theta) = (-810 + 2430Sin(theta)) / -135

Step 6: Simplify further
To simplify further, we can multiply both sides by -135 to get rid of the negative sign on the right side.

-135 / Cos(theta) = -810 + 2430Sin(theta)

Step 7: Rearrange the equation
Rearrange the equation to isolate the Sin(theta) term.

2430Sin(theta) = -135 / Cos(theta) + 810

Step 8: Simplify further
Divide both sides by 2430 to solve for Sin(theta).

Sin(theta) = (-135 / Cos(theta) + 810) / 2430

Step 9: Simplify the expression inside the bracket
Simplify the expression inside the bracket by dividing both terms by 135.

Sin(theta) = (-1 / (Cos(theta) / 135) + 6) / 2430

Step 10: Simplify further
To simplify further, multiply both the numerator and denominator by 135 to get rid of the fraction inside the bracket.

Sin(theta) = (-1 / Cos(theta) + 6*135) / (2430 * 135)

Step 11: Simplify the expression inside the bracket
Simplify the expression inside the bracket by multiplying 6 and 135.

Sin(theta) = (-1 / Cos(theta) + 810) / (328050)

Step 12: Simplify further
To simplify further, multiply both the numerator and denominator by Cos(theta) to eliminate the fraction.

Sin(theta)Cos(theta) = (-1 + 810Cos(theta)) / (328050)

Step 13: Rearrange the equation
Rearrange the equation to isolate the Theta term.

-810Cos(theta) + 2430Sin(theta)Cos(theta) = -135

2430Sin(theta)Cos(theta) = 810Cos(theta) - 135

Step 14: Substitute Sin(theta)Cos(theta) in terms of Cos(theta)
Substitute Sin(theta)Cos(theta) with (-1 + 810Cos(theta)) / (328050) into the equation.

(-1 + 810Cos(theta)) / (328050) = 810Cos(theta) - 135

Step 15: Solve for Cos(theta)
Multiply both sides of the equation by (328050).

-1 + 810Cos(theta) = 328050(810Cos(theta)) - 135(328050)

Simplify and solve the equation to find the value of Cos(theta).

After finding the value of Cos(theta), you can substitute it back into any of the previous equations to find the value of Sin(theta) or theta itself, depending on what you are looking for.

To find the value of theta, you can simplify the given equation step by step. Let's go through the steps together:

1. Start with the given equation:
-810Cos(theta) + 135 + 2430 Sin(theta)Cos(theta) = 0

2. Combine like terms:
2430 Sin(theta)Cos(theta) - 810Cos(theta) + 135 = 0

3. Factor out common terms:
Cos(theta) (2430 Sin(theta) - 810) + 135 = 0

4. Divide through by (2430 Sin(theta) - 810):
Cos(theta) = -135 / (2430 Sin(theta) - 810)

5. Simplify the expression:
Cos(theta) = -135 / (2430 Sin(theta) - 810)
= -135 / 270(9 Sin(theta) - 3)

6. Simplify further:
Cos(theta) = -1 / (2(9 Sin(theta) - 3))
= -1 / (18 Sin(theta) - 6)

7. Take the reciprocal of both sides:
1 / Cos(theta) = 18 Sin(theta) - 6

8. Rewrite as a trigonometric identity:
Sec(theta) = 18 Sin(theta) - 6

At this point, we have obtained a simplified equation in terms of secant and sine. To find the value of theta, you can use a calculator or trigonometric identities to solve this equation.