Question

in right triangle xyz, ∠x and ∠z are complementary angles and cos(x) is 9/11. what is sin(z)?
a. √20/11
b. 11/9
c. 9/11
d. √20/9

Explanation:

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

In a right - triangle \(XYZ\) with \(\angle Y = 90^{\circ}\), \(\angle X+\angle Z=90^{\circ}\), so \(\angle Z = 90^{\circ}-\angle X\).
We know the trigonometric identity \(\sin(A)=\cos(90^{\circ}-A)\).
Let \(A = Z\), then \(\sin(Z)=\cos(90^{\circ}-Z)\). Since \(\angle X = 90^{\circ}-\angle Z\), we have \(\sin(Z)=\cos(X)\).

Step2: Substitute the given value of \(\cos(X)\)

Given \(\cos(X)=\frac{9}{11}\).

Answer:

C. \(\frac{9}{11}\)