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  4. If sin θ+ cos θ=(1/5) and 0 ≤ θ<π, then tan θ is

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Q. If $\sin \theta+\cos \theta=\frac{1}{5}$ and $0 \leq \theta<\pi$, then $\tan \theta$ is

Trigonometric Functions

Solution:

$\cos \theta+\sin \theta=\frac{1}{5}$
$\Rightarrow 1+\tan \theta=\frac{1}{5} \sec \theta$ (dividing by $\cos \theta$ )
$\Rightarrow 25+50 \tan \theta+25 \tan ^{2} \theta=\sec ^{2} \theta$
or $12 \tan ^{2} \theta+25 \tan \theta+12=0$
or $12 \tan ^{2} \theta+16 \tan \theta+9 \tan \theta+12=0$
or $(4 \tan \theta+3)(3 \tan \theta+4)=0$
$\Rightarrow \tan \theta=-\frac{3}{4}$
or $\tan \theta=-\frac{4}{3}$
$\tan \theta=-\frac{3}{4}$ is rejected as then $\cos \theta=-\frac{4}{5}$ and $\sin \theta=\frac{3}{5}$
for which $\cos \theta+\sin \theta=-\frac{1}{5}$
Hence, $\tan \theta=-\frac{4}{3}$