Question

In $\Delta ABC$, right angled at $B$, if one angle is ${45}^{\circ }$, find the value of $\sin A,\cos C,\cot A$ and $\tan C$ respectively.

• A. $\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},1,1$
• B. $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},1,1$
• C. $2,2,\sqrt{2},\sqrt{2}$
• D. $1,1,\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},1,1$



In right angled $\Delta ABC$,
Let $\angle A={45}^{\circ },$ $\angle B={90}^{\circ }$
Then $\angle C={180}^{\circ }-{90}^{\circ }-{45}^{\circ }={45}^{\circ }$
Hence triangle is right isosceles triangle with $AB=BC=a\left(say\right)$
Apply Pythagoras theorem in $\Delta ABC$
$AC=\sqrt{(a^2+a^2)}=a\sqrt{2}$
Using Trigonometric ratio
$\sin A=\frac{BC}{AC}=\frac{a}{a\sqrt{2}}=\frac{1}{\sqrt{2}}$
$\cos C=\frac{BC}{AC}=\frac{a}{a\sqrt{2}}=\frac{1}{\sqrt{2}}\cot A=\frac{AB}{BC}=\frac{a}{a}=1\tan C=\frac{AB}{BC}=\frac{a}{a}=1$