Find all solutions of the equation Tan(theta - 30 degrees)= -5 in interval 0 - 360 Book gives answers for theta of 131.3 and 311.3

To find all solutions of the equation Tan(theta - 30 degrees) = -5 in the interval 0 - 360, we can follow these steps:

Step 1: Convert the given equation from degrees to radians.
Tan(theta - 30 degrees) = -5
Convert 30 degrees to radians: 30 degrees * pi/180 = pi/6
So the equation becomes Tan(theta - pi/6) = -5

Step 2: Solve the equation for theta.
Start by taking the arctangent of both sides to isolate the angle:
Theta - pi/6 = arctan(-5)
Theta = arctan(-5) + pi/6

Step 3: Calculate the two solutions for theta in the specified interval.
- Use a calculator or a table of tangent values to find the arctan(-5) value.
The calculator will give you an angle in radians, so convert it to degrees.
- Add pi/6 to the angle found in the previous step to get the first solution for theta.
- Subtract pi from the angle found in the previous step, then add pi/6 to get the second solution for theta.

Let's do the calculations:

Step 3 (cont'd): Calculate the arc tangent of -5:
arctan(-5) ≈ -78.69 degrees

Step 3 (cont'd): Convert the angle from degrees to radians:
-78.69 degrees * pi/180 ≈ -1.37 radians

Step 3 (cont'd): Calculate the first solution for theta:
Theta1 ≈ (-1.37 radians) + (pi/6) ≈ -0.57 radians ≈ -32.69 degrees

Step 3 (cont'd): Calculate the second solution for theta:
Theta2 ≈ (-1.37 radians) + pi - (pi/6) ≈ 4.71 radians ≈ 269.31 degrees

Therefore, the solutions for theta in the interval 0 - 360 are approximately -32.69 degrees (or -0.57 radians) and 269.31 degrees (or 4.71 radians).

tan(Ø - 30) = -5

we know tan 78.69° = +5
and since the tangent of our given angle of Ø-30 is negative it must be quadrants II or IV

Ø-30 = 180-78.69 = 101.31
Ø = 101.31+30 = 131.3°
ir
Ø-30 = 360 - 78.69 = 281.3
Ø = 311.3°

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