Question

in right triangle abc,m∠a = 30° and m∠b = 90°. 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. i and ii b. ii and iv c. i and iii d. iii and iv

Explanation:

Step1: Find angle C

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

Step2: Calculate sine and cosine values

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

Step3: Check each statement

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

eq\sin(C)$.

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

eq\cos(C)$.

• For IV: $\cos(A)=\frac{\sqrt{3}}{2}$ and $\sin(C)=\frac{\sqrt{3}}{2}$, so $\cos(A)=\sin(C)$.

Answer:

B. II and IV