Question
If $\sin\theta=\frac{1}{\sqrt{2}},$ find all other trigonometric rations of angle $\theta$.
✓
Answer
We have $\sin\theta=\frac{1}{\sqrt{2}}$
In $\triangle\text{ABC}$
$\text{AC}^2=\text{AB}^2+\text{BC}^2$
$\Rightarrow\ (\sqrt{2})=(1)^2+\text{BC}^2$
$\Rightarrow\ \text{BC}^2=2-1$
$\Rightarrow\ \text{BC}=1$
$\therefore\ \cos\theta=\frac{\text{BC}}{\text{AC}}=\frac{1}{\sqrt{2}}$
$\tan\theta=\frac{\text{AB}}{\text{BC}}=\frac{1}{1}=1$
$\cot\theta=\frac{1}{\tan\theta}=1$
$\sec\theta=\frac{1}{\cos\theta}=\sqrt{2}$
$\text{cosec }\theta=\frac{1}{\sin\theta}=\sqrt{2}$
In $\triangle\text{ABC}$
$\text{AC}^2=\text{AB}^2+\text{BC}^2$
$\Rightarrow\ (\sqrt{2})=(1)^2+\text{BC}^2$
$\Rightarrow\ \text{BC}^2=2-1$
$\Rightarrow\ \text{BC}=1$
$\therefore\ \cos\theta=\frac{\text{BC}}{\text{AC}}=\frac{1}{\sqrt{2}}$
$\tan\theta=\frac{\text{AB}}{\text{BC}}=\frac{1}{1}=1$
$\cot\theta=\frac{1}{\tan\theta}=1$
$\sec\theta=\frac{1}{\cos\theta}=\sqrt{2}$
$\text{cosec }\theta=\frac{1}{\sin\theta}=\sqrt{2}$
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