Question

select the true statement about triangle abc.
a. $\cos a = \sin b$
b. $\cos a = \cos c$
c. $\cos a = \sin c$
d. $\cos a = \tan c$

Explanation:

Step1: Recall trigonometric ratios

For right triangle $ABC$ (right-angled at $B$):

• $\cos A = \frac{\text{adjacent to } A}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{12}{13}$
• $\sin B = \sin 90^\circ = 1$
• $\cos C = \frac{\text{adjacent to } C}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{5}{13}$
• $\sin C = \frac{\text{opposite to } C}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{12}{13}$
• $\tan C = \frac{\text{opposite to } C}{\text{adjacent to } C} = \frac{AB}{BC} = \frac{12}{5}$

Step2: Match values to options

Compare $\cos A = \frac{12}{13}$ with each option's right-hand side:

• Option A: $\sin B = 1

eq \frac{12}{13}$

• Option B: $\cos C = \frac{5}{13}

eq \frac{12}{13}$

• Option C: $\sin C = \frac{12}{13} = \cos A$
• Option D: $\tan C = \frac{12}{5}

eq \frac{12}{13}$

Answer:

C. $\cos A = \sin C$