Sine and Cosine question

**Mathsaholic** · April 28th 2010, 01:29 AM

I am learning sine, cosine and trigonometry graphs, and other trig stuff. And I am really stuck with this question:

1) What is the value of x???

2sinx + cosx = 1.5 (x is between 0 and 180)

I would really appreciate it if someone could help me with this question.
Thank You ^^

**Failure** · April 28th 2010, 02:30 AM

Originally Posted by Mathsaholic

I am learning sine, cosine and trigonometry graphs, and other trig stuff. And I am really stuck with this question:

1) What is the value of x???

2sinx + cosx = 1.5 (x is between 0 and 180)

I would really appreciate it if someone could help me with this question.
Thank You ^^

If you have to solve an equation of the form $a\sin x+b\cos x=c$ , you can divide the whole equation by $\sqrt{a^2+b^2}$ . The new coefficients of $\sin x$ and $\cos x$ have the property that the sum of their squares is 1. This allows you to apply the Addition Formula for the $\sin$ , which says that $\sin(x)=\sin(x)\cos(x_0)+\cos(x)\sin(x_0)$ .
Then you use the two equations $\cos(x_0)=a/\sqrt{a^2+b^2}$ and $\sin(x_0)=b/\sqrt{a^2+b^2}$ , to figure out what $x_0$ is.

So in your case this works as follows
$2\sin(x)+\cos(x)=1.5$
$\frac{2}{\sqrt{5}}\sin(x)+\frac{1}{\sqrt{5}}\cos(x )=\frac{1.5}{\sqrt{5}}$
$\sin(x+x_0)=\frac{1.5}{\sqrt{5}}$
Now you apply $\sin^{-1}$ to both sides to get the general solution
$x+x_0\approx \begin{cases}42.13^\circ+n\cdot360^\circ\\137.87^\ circ+n\cdot 360^\circ\end{cases}, n\in \mathbb{Z}$

Finally you have that $\cos(x_0)=\frac{2}{\sqrt{5}}$ and $\sin(x_0)=\frac{1}{\sqrt{5}}$ , in other words $\tan(x_0)=\sin(x_0)/\cos(x_0)=\frac{1}{2}$ , which means, $x_0\approx 26.57^\circ$ (we don't need the general solution for $x_0$ here).

Thus you have

$x\approx \begin{cases}42.13^\circ-26.57^\circ+n\cdot360^\circ\\137.87^\circ-26.57^\circ+n\cdot 360^\circ\end{cases}, n\in \mathbb{Z}$
Now, from these infinitely many solutions you need to pick those, that lie in the reqired intervall.

Thread: Sine and Cosine question

LinkBack

Thread Tools

Display

Sine and Cosine question

Similar Math Help Forum Discussions

Question about sine and cosine law

sine and cosine

Sine and Cosine rules question

A question based on the sine & cosine formulas

Cosine/Sine

Search Tags