Solving trigonometrical equations with the R formula

Let’s take a dig at trigonometry.

Solving a trigonometric equation in A Maths has always been equating a trigonometric function to a value. For example, for the equation

                sin x = 0.5

we can ascertain the value of x by finding arc sin 0.5  (sin^-1 0.5).

Even if it were a little more complex, like

                tan²x = 0.25

we can still find the values for x since there is only one function on the end of the equation.

But what happens when there are two functions? Well, that means we need to use our identities.

For example, if

                sin²x + 2cos²x = 1

then we will need to either convert sin²x to cos²x or vice versa. This is using the identity

                sin²x + cos²x = 1

Still, no problem.

But what happens when we do not have an apparent identity to link the trigonometric functions together? Something like       2 sin x – cos x = 0

Well, the trick here is to move the cos x to the other end of the equation, like so:

                2 sin x =  cos x

and then by dividing throughout by sin x we will get a tan x function. So, still a single function.

The problem above becomes a little tricky when instead of  2 sin x – cos x = 0, we had  2 sin x – cos x = 1.  Now we cannot use this trick. We need another identity. And that is the R formula.

Let’s solve the question above

2 sin x – cos x = 1    for  0 ≤ x ≤ 2π      (Note that the R formula can only be used for angles in radian)

2 sin x – cos x ≡ R sin (x – α), where R = √(2² + 1²) = √5 

                                                and tan α= ½ ;   α = tan^-1( ½) = 0.464

Therefore,    

2 sin x – cos x = √5sin(x – 0.464) =  1

sin(x – 0.464) = 1/√5  = 0.4472

x – 0.464 = sin^-10.4472 = 0.464,  2.68

Hence,  x = 0.464 + 0.464    or   x = 2.68 + 0.464

                = 0.928                            = 3.144

Therefore  x = 0.928 , 3.144.

 And that, as they say, is that!