$$3\sin x + 4\cos x = 5$$ then the value of $\tan(x/2)$
I should get the answer as $1/3$ but my answer is $(5\sec x-4)/3$.
Can any one help me thanks in advance.
Asked
Active
Viewed 66 times
-1
-
Am trying to to get the tanx from RHS... Can you give me the solution please? – Pallavi Konda Aug 15 '15 at 01:54
-
Check out this youtube video: https://www.youtube.com/watch?v=ijsUvXPdMxE This is what Euler88 is referring too – imranfat Aug 15 '15 at 02:15
-
The way to proceed is almost in the question. Use the tangent half-angle substitution $t=\tan(\frac x2)$ just as Zhanxiong answered. – Claude Leibovici Aug 15 '15 at 05:16
-
This is rather similar to: How to solve $4\sin \theta +3\cos \theta = 5$ – Martin Sleziak Apr 20 '18 at 13:59
2 Answers
2
Let $t = \tan(x/2)$, apply the celebrated formula: $$\sin x = \frac{2\tan(x/2)}{1 + \tan^2(x/2)} = \frac{2t}{1 + t^2}, \quad \cos x = \frac{1 - \tan^2(x/2)}{1 + \tan^2(x/2)} = \frac{1 - t^2}{1 + t^2}.$$ Plug these back into the given equation, we have $$\frac{6t}{1 + t^2} + \frac{4(1 - t^2)}{1 + t^2} = 5.$$
This is a standard quadratic formula, can you take it from here?
Zhanxiong
- 14,040
-
-
Then how about @Euler88 solution. It needs sharper observation though. – Zhanxiong Aug 15 '15 at 02:07
-
0
Hint: $\sin^2x+\cos^2x=1$ and $3\sin x+4\cos x=5$ implies $\sin x=3/5$ and $\cos x=4/5$. So you can conclude.
Euler88 ...
- 2,090
-
@Hello, How have you concluded $\tan\dfrac x2$ from the answer? – lab bhattacharjee Aug 15 '15 at 02:56
-
Using that $\tan(x-y)=\frac{\tan(x)-\tan(y)}{1+\tan(x)\tan(y)}$. Change $y=x/2$ and conclude. – Euler88 ... Aug 15 '15 at 06:19