# Thread: Trigonometric and Hyperbolic identities

1. ## Trigonometric and Hyperbolic identities

Question is as follows:-

Hence solve for t for values in the range 0 ≤ t ≤ 2 π rad:
5.5 Cos t + 7.8 Sin t = 4.5

Any help would be greatful, thanks

2. ## Re: Trigonometric and Hyperbolic identities

Originally Posted by boza100
Question is as follows:-

Hence solve for t for values in the range 0 ≤ t ≤ 2 π rad:
5.5 Cos t + 7.8 Sin t = 4.5

Any help would be greatful, thanks
Well it's a bit ugly but one approach would be to recall that $cos(t) = \sqrt{1 - sin^2(t)}$ and plug that in...
$5.5~\sqrt{1 - sin^2(t)} + 7.8~sin(t) = 4.5$

Isolate the radical and square it. This will give you a quadratic in sin(t) which you can solve using the quadratic formula. Check for extra, but not valid, solutions.

Also since in reality $cos(t) = \pm \sqrt{1 - sin^2(t)}$ you should also work through the negative solution as well. And again check your solutions with the original equation.

-Dan

3. ## Re: Trigonometric and Hyperbolic identities

Another approach would be to use a linear combination identity to write the equation as:

$\sqrt{5.5^2+7.8^2}\sin\left(t+\tan^{-1}\left(\frac{5.5}{7.8} \right) \right)=4.5$

$\sin\left(t+\tan^{-1}\left(\frac{55}{78} \right) \right)=\frac{45}{\sqrt{9109}}$

Now, after finding the quadrant IV solution, use the identity $\sin(\pi-x)=\sin(x)$ to get the quadrant III solution.