Question

How to write $2\sin \theta -4\cos \theta $ in the form $r\sin \left( \theta -\alpha \right)$ ?
(a) Using linear formulas
(b) Using trigonometric identities
(c) Using algebraic properties
(d) None of these

Answer 1

Hint: To start with, in this problem we are to write $2\sin \theta -4\cos \theta $ in the form of $r\sin \left( \theta -\alpha \right)$. We can do it by expanding the value $\sin \left( \theta -\alpha \right)$ and compare it with the given values. Then simplifying the values we will get our needed solution.

Complete step by step solution:
According to the question, we are to write $2\sin \theta -4\cos \theta $ in the given form of $r\sin \left( \theta -\alpha \right)$.
We will start with explaining the given formula, $r\sin \left( \theta -\alpha \right)$as $r\sin \theta \cos \alpha -r\cos \theta \sin \alpha $.
Now, we can try to equate both equations to find the solution.
So, we can compare the coefficients of $\sin \theta $ and $\cos \theta $ and write them altogether.
Then, we get, $r\cos \alpha =2$ and $r\sin \alpha =4$ .
Now, to find the value r, we will square both of them and add them together.
So, we get, ${{r}^{2}}{{\cos }^{2}}\alpha =4$and ${{r}^{2}}{{\sin }^{2}}\alpha =16$
Thus, adding them, ${{r}^{2}}{{\sin }^{2}}\alpha +{{r}^{2}}{{\cos }^{2}}\alpha =4+16=20$
From the trigonometric identities, ${{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha =1$.
Taking ${{r}^{2}}$common from the left hand side, now we have, ${{r}^{2}}\left( {{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha \right)=20$
Using the identity, we have, ${{r}^{2}}=20$
And from this we have, $r=\sqrt{20}=2\sqrt{5}$
Again, dividing $r\sin \alpha $by $r\cos \alpha $now, we get, $\dfrac{r\sin \alpha }{r\cos \alpha }=\tan \alpha $ which is having a value 2.
So, we have the value of $\alpha $as, $\alpha ={{\tan }^{-1}}2$(using the trigonometric inverse identities).
Thus, we can put the values in $r\sin \left( \theta -\alpha \right)$to get our solution.
Putting back the values, we can write $2\sin \theta -4\cos \theta $as $2\sqrt{5}\sin \left( \theta -{{\tan }^{-1}}2 \right)$ .

So, the correct answer is “Option (b)”.

Note: In this problem, we have used the trigonometric angle identities to simplify and get the solution. In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.