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# On Solving Trigonometric Equations 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.

How to Solve a Trigonometric Equation ~ Example #1 ~ tany - 2 = coty

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.

How to Solve a Trigonometric Equation ~ Example #2 ~ 2sin(x + π/3) = -1

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 don’t know if you remember me, but you really helped me last year and I just wanted to tell you I came first in my maths half yearly with 100%. A lot of it goes to you. Thanks for your help.
Jashan M (student, 2013)

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