Answer:
Therefore,
[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]
[tex]\cot B =\dfrac{1}{2}[/tex]
[tex]\csc B = \dfrac{\sqrt{5}}{2}[/tex]
Step-by-step explanation:
Given:
Right △ABC has its right angle at C,
BC=4 , and AC=8 .
To Find:
Cos A = ?
Cot B = ?
Csc B = ?
Solution:
Right △ABC has its right angle at C, Then by Pythagoras theorem we have
[tex](\textrm{Hypotenuse})^{2} = (\textrm{Shorter leg})^{2}+(\textrm{Longer leg})^{2}[/tex]
Substituting the values we get
[tex](AB)^{2}=4^{2}+8^{2}=80\\AB=\sqrt{80}\\AB=4\sqrt{5}[/tex]
Now by Cosine identity
[tex]\cos A = \dfrac{\textrm{side adjacent to angle A}}{Hypotenuse}\\[/tex]
Substituting the values we get
[tex]\cos A = \dfrac{AC}{AB}=\dfrac{8}{4\sqrt{5}}=\dfrac{2}{\sqrt{5}}\\\\Ratinalizing\\\cos A=\dfrac{2\sqrt{5}}{5}[/tex]
[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]
Now by Cot identity
[tex]\cot B = \dfrac{\textrm{side adjacent to angle B}}{\textrm{side opposite to angle B}}[/tex]
Substituting the values we get
[tex]\cot B = \dfrac{BC}{AC}=\dfrac{4}{8}=\dfrac{1}{2}[/tex]
Now by Cosec identity
[tex]\csc B = \dfrac{Hypotenuse}{\textrm{side opposite to angle B}}\\[/tex]
Substituting the values we get
[tex]\csc B = \dfrac{AB}{AC}=\dfrac{4\sqrt{5}}{8}=\dfrac{\sqrt{5}}{2}[/tex]
Therefore,
[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]
[tex]\cot B =\dfrac{1}{2}[/tex]
[tex]\csc B = \dfrac{\sqrt{5}}{2}[/tex]
