# Solving trigonometry equations from graphs when transformations (e.g. $3\tan^2$) are involved?

Preface; I'm not asking for answers (that won't help me learn Maths at all!), but certainly some guidance would be very much appreciated.

I'm trying to solve some trigonometry equations from graphs, and it's all going fine; until transformations are involved.

I'm aware that a number before the $\tan,\sin,\cos$ (e.g. $3\cos$) 'compresses' the graph by a scale factor of said number. But what if there is a squared sign infront of it as well?

Example equations which I'm struggling with -
$$\cos^2(x) = 3/4$$ $$3\tan^2(x) = 1$$

Certainly with the first one, I'm in a mess already because 3/4 doesn't appear in the common trignometry values (http://www.analyzemath.com/trigonometry/special_angles_1.gif).

Cameron.

• I'm not sure, but are you aware that $\cos^2(x)$ means $(\cos(x))^2$, and likewise for other trigonomatric functions? – Herng Yi Jan 20 '13 at 12:01

If your question appears in exercises that do not require the use of a calculator, it is almost certain that only special angles will be needed. For the two cases you mentioned, simply manipulate the equations until you get the familiar "trigo function = something" and solve it.

For instance,

1. suppose that $\cos^2(x) = 3/4$.
2. Let $C = \cos(x)$; the equation then becomes $C^2 = 3/4$.
3. Solve for $C$; $C = -\sqrt{3}/2$ or $\sqrt{3}/2$.
4. Look at the equation again: $\cos(x) = -\sqrt{3}/2$ or $\sqrt{3}/2$.

So, simply "peel off" everything from the trigonometric function and make it the subject, then see if the other side of the equation corresponds to a special angle.

If $\cos^2(x) = 3/4$, then $\cos(x) = \pm\sqrt{3}/2$, which corresponds to a special angle.

If $3\tan^2(x) = 1$, then $\tan(x) = \pm1/\sqrt 3$, which corresponds to the same special angle.