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) = 28/29, the value of cos(c) =
if cos(c) = 28/29, the value of sin(a) =
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Explanation:

Step1: Use the property of complementary angles in a right - triangle

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

Step2: Find the values

If \(\sin(A)=\frac{3}{8}\), then \(\cos(C)=\sin(A)\) (because \(A + C = 90^{\circ}\), so \(C=90 - A\) and \(\cos(C)=\cos(90 - A)=\sin(A)\)).
If \(\cos(C)=\frac{5}{8}\), then \(\sin(A)=\cos(C)\) (because \(A=90 - C\) and \(\sin(A)=\sin(90 - C)=\cos(C)\))

Answer:

If \(\cos(C)=\frac{3}{8}\), then \(\sin(A)=\frac{3}{8}\); if \(\sin(A)=\frac{5}{8}\), then \(\cos(C)=\frac{5}{8}\)