Question

find the lengths of the missing sides if side ( a ) is opposite angle ( a ), side ( b ) is opposite angle ( b ), and side ( c ) is the hypotenuse.

( c = 10, angle a = 45 ^ { circ } )

( a = )

( b = )

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Explanation:

Step1: Find side \(a\)

Use the sine function: \(\sin A=\frac{a}{c}\).
Given \(c = 10\) and \(A = 45^{\circ}\), \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\).
So, \(a=c\times\sin A=10\times\frac{\sqrt{2}}{2}=5\sqrt{2}\).

Step2: Find side \(b\)

Use the cosine function: \(\cos A=\frac{b}{c}\).
Given \(c = 10\) and \(A = 45^{\circ}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\).
So, \(b=c\times\cos A=10\times\frac{\sqrt{2}}{2}=5\sqrt{2}\).

Answer:

\(a = 5\sqrt{2}\), \(b = 5\sqrt{2}\)