a.\(sin\widehat{A}=\sqrt{1-\left(cos\widehat{A}\right)^2}=\dfrac{1}{\sqrt{2}}\)
\(tan\widehat{A}=\dfrac{sin\widehat{A}}{cos\widehat{A}}=1, cot\widehat{A}=\dfrac{1}{tan\widehat{A}}=1\)
b.\(\dfrac{1}{\left(cos\widehat{A}\right)^2}=\left(tan\widehat{A}\right)^2+1 \Leftrightarrow cos\widehat{A}=\sqrt{\dfrac{1}{\left(tan\widehat{A}\right)^2+1}}=\dfrac{1}{2}\)
\(sin\widehat{A}=\sqrt{1-\left(cos\widehat{A}\right)^2}=\dfrac{\sqrt{3}}{2}\), \(cot\widehat{A}=\dfrac{1}{tan\widehat{A}}=\dfrac{1}{\sqrt{3}}\)
c.\( cos\widehat{A}=\sqrt{1-\left(sin\widehat{A}\right)^2}=\dfrac{\sqrt{21}}{5}, tan\widehat{A}=\dfrac{sin\widehat{A}}{cos\widehat{A}}\)\(=\dfrac{2\sqrt{21}}{21}\),\(cot\widehat{A}=\dfrac{\sqrt{21}}{2}\)
d.\(\dfrac{1}{\left(sin\widehat{A}\right)^2}=\left(cot\widehat{A}\right)^2+1\)\(\Leftrightarrow sin\widehat{A}=\dfrac{4}{5}\), \(cos\widehat{A}=\sqrt{1-\left(sin\widehat{A}\right)^2}=\dfrac{3}{5}\),\(tan\widehat{A}=\dfrac{1}{cot\widehat{A}}=\dfrac{4}{3}\)
