All equations are solved using inverse functions. That is, we “undo” operations on a variable to ultimately find the value of that variable.

For example, When we solve 2x + 3 = 15, we first subtract 3 from both sides of the equation. This is because -3 is the inverse operation to +3, and this allows us to remove the + 3 from the left hand side of the equation. We now have 2x = 12. Now, notice that 2x means that x has been multiplied by two. The inverse operation to multiplying by two is dividing by two (÷2 or /2), so we divide both sides of the equation by two and obtain our “solution,” x = 6!

In a similar way, when we need to solve sin(x) = 0.5, for example, we need some function or operation to “remove the sine function from the x” on the left hand side of the equation. Mathematicians devised such a function and called it the “inverse sine” function. It can be written sin‾¹ (or arcsin in the USA) and is called “the inverse sine” function. Therefore, our solution should look similar to that in the image above.

There are more “tricks to the trade” as the complexity of equations increases, but the priciple that I have shown you here underlies them all.

One of my YouTube subscribers presented me with two trigonometric equations to solve. This is the first of them:

tany - 2 = coty, where 0° ≤ y ≤ 180°

If you watch this video you should discover:

key things to look for in such an equation,

how to resolve it into a quadratic equation,

how to check the nature of the roots/zeros

how to find the roots/zeros,

how to use the unit circle to find the relevant angles, and

how to identify the angles within the restricted domain.

The video is a little longer than it might have been because I explain each step without skipping over too many details. If you have difficulty solving equations of this type you should learn some useful skills here.

This is the second of two trigonometric equations that one of my YouTube subscribers asked me to solve.

The equation in question is:

2.sin(x + π/3) = -1, where 0 ≤ x ≤ 2π

If you watch this video, you should discover:

key things to look for in such an equation,

how to simplify the equation,

how to use triangles and the unit circle to identify angles, and

how to identify the angles within the restricted domain.

I enjoyed your presentation and no it wasn’t too long. Each subtraction algorithm has its merit as you demostrated, but after learning the “one up and one down” method, I’m employing it because of its speed and ease of usage. Even my wife, who hates mathematics with a passion, thinks it’s too easy. I look forward to your future presentations on both multiplication and number theory. I read an introduction text book some twenty five years ago on number theory by Oystein Ore who taught at Yale for better than twenty years. So in closing, please produce these lectures and the longer the better. Thanks.
Dennis Bell (on a CCM YouTube video about How to Subtract (Large) Numbers Easily)