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How do I solve this type of problem?
Without the use of a calculator, find all values of theta from [0,2pi).
tan(theta)=5
sec(theta)= -9/5
I have a ton of these problems but don't understand how to do any of them. If someone could explain how to do them, that would be fantastic.
1 Answer
- ValithorLv 41 decade agoFavorite Answer
The first thing to realise is that equations of the forms: sin(x) = a, cos(x) = a, tan(x) = a, always have two solutions in the domain [0, 2π).
Note: equations of the form sin(bx) = a, cos(bx) = a, tan(bx) = a, can have more or less solutions.
Your calculator will typically provide only the solution closest to zero (x1).
Eg:
tan(x) = 5
x = arctan(5)
x1 = 1.3734
(arctan is the inverse function to tan, on a calculator it will appear tan^-1)
To find the second solution (x2) in the domain [0, 2π), use the following rules:
For sin: x2 = π - x1
For cos: x2 = 2π - x1
For tan: x2 = π + x1
Eg:
tan(x) = 5
x = arctan(5)
x1 = 1.3734
x2 = 4.51499
For periodic functions of the form: y = sec(x), y = csc(x), y = cot(x), start by changing them to sin(x), cos(x) or tan(x).
Eg:
sec(x) = -9/5
1/cos(x) = -9/5
cos(x) = -5/9
x = arccos(-5/9)
x1 = 2.15983
x2 = 4.12336
Note: all my solutions are in radian measure. If you want solutions in degrees, replace π with 180º in the calculations.
Hope that helped :)