Question

type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s).
a right triangle abc has complementary angles a and c.
if sin(a) = 24/25, the value of cos(c) =
if cos(c) = 20/29, the value of sin(a) =

Explanation:

Step1: Use the co - function identity

In a right - triangle \(ABC\) with \(\angle A+\angle C = 90^{\circ}\), we know the co - function identity \(\sin(A)=\cos(90^{\circ}-A)\) and \(\cos(C)=\sin(90^{\circ}-C)\). Since \(A + C=90^{\circ}\), then \(\sin(A)=\cos(C)\) and \(\cos(A)=\sin(C)\)

Step2: Find the value of \(\cos(C)\) when \(\sin(A)=\frac{24}{25}\)

Because \(A + C = 90^{\circ}\), by the co - function identity \(\cos(C)=\sin(A)\). Given \(\sin(A)=\frac{24}{25}\), so \(\cos(C)=\frac{24}{25}\)

Step3: Find the value of \(\sin(A)\) when \(\cos(C)=\frac{20}{29}\)

Since \(A + C=90^{\circ}\), by the co - function identity \(\sin(A)=\cos(C)\). Given \(\cos(C)=\frac{20}{29}\), so \(\sin(A)=\frac{20}{29}\)

Answer:

\(\frac{20}{29},\frac{24}{25}\)