Question

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

Explanation:

Step1: Calculate \(\cos A\)

Using the definition of cosine (\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)) in right - triangle \(ABC\) for angle \(A\).
The adjacent side to angle \(A\) is \(AB = 12\) and the hypotenuse \(AC=13\). So, \(\cos A=\frac{AB}{AC}=\frac{12}{13}\)

Step2: Calculate \(\tan C\)

Using the definition of tangent (\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)) in right - triangle \(ABC\) for angle \(C\).
The opposite side to angle \(C\) is \(AB = 12\) and the adjacent side to angle \(C\) is \(BC = 5\). So, \(\tan C=\frac{AB}{BC}=\frac{12}{5}\)

Step3: Calculate \(\sin B\)

Since \(\angle B = 90^{\circ}\), \(\sin B=\sin90^{\circ}=1\)

Step4: Calculate \(\sin C\)

Using the definition of sine (\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)) in right - triangle \(ABC\) for angle \(C\).
The opposite side to angle \(C\) is \(AB = 12\) and the hypotenuse \(AC = 13\). So, \(\sin C=\frac{AB}{AC}=\frac{12}{13}\)

Step5: Calculate \(\cos C\)

Using the definition of cosine (\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)) in right - triangle \(ABC\) for angle \(C\).
The adjacent side to angle \(C\) is \(BC = 5\) and the hypotenuse \(AC=13\). So, \(\cos C=\frac{BC}{AC}=\frac{5}{13}\)

Answer:

D. \(\cos A=\sin C\)