Question

in right triangle abc, ( mangle a = 30^{circ} ) and ( mangle b = 90^{circ} ). which of the following are true?
i. ( sin(a)=sin(c) )
ii. ( sin(a)=cos(c) )
iii. ( cos(a)=cos(c) )
iv. ( cos(a)=sin(c) )
a. iii and iv
b. i and ii
c. ii and iv
d. i and iii

Explanation:

Step1: Find the measure of angle \(C\)

In a triangle, the sum of angles is \(180^{\circ}\). Given \(m\angle A = 30^{\circ}\) and \(m\angle B=90^{\circ}\), then \(m\angle C=180^{\circ}-(90^{\circ} + 30^{\circ})=60^{\circ}\).

Step2: Calculate \(\sin(A)\), \(\sin(C)\), \(\cos(A)\) and \(\cos(C)\)

• \(\sin(A)=\sin(30^{\circ})=\frac{1}{2}\)
• \(\sin(C)=\sin(60^{\circ})=\frac{\sqrt{3}}{2}\)
• \(\cos(A)=\cos(30^{\circ})=\frac{\sqrt{3}}{2}\)
• \(\cos(C)=\cos(60^{\circ})=\frac{1}{2}\)

Step3: Check each statement

• Statement I: \(\sin(A)=\frac{1}{2}\), \(\sin(C)=\frac{\sqrt{3}}{2}\), so \(\sin(A)

eq\sin(C)\)

• Statement II: \(\sin(A)=\frac{1}{2}\), \(\cos(C)=\frac{1}{2}\), so \(\sin(A)=\cos(C)\)
• Statement III: \(\cos(A)=\frac{\sqrt{3}}{2}\), \(\cos(C)=\frac{1}{2}\), so \(\cos(A)

eq\cos(C)\)

• Statement IV: \(\cos(A)=\frac{\sqrt{3}}{2}\), \(\sin(C)=\frac{\sqrt{3}}{2}\), so \(\cos(A)=\sin(C)\)

Answer:

C. II and IV