Discussion
In a right angled triangle ΔABC ∠B = 90° and \(\sin A = 5/\sqrt{29}\). Then which of the following is not true?
Answer: D
The statement that is not true is \(\sec C = \sqrt{29}/2\).
With the right angle at B, \(\sin A = BC/AC = 5/\sqrt{29}\). Take BC = 5 and AC = \(\sqrt{29}\); then
\[ AB = \sqrt{29 - 25} = 2 \]
So \(\cos A = 2/\sqrt{29}\), \(\tan C = AB/BC = 2/5\) and \(\cos C = BC/AC = 5/\sqrt{29}\), all true. But \(\sec C = AC/BC = \sqrt{29}/5\), not \(\sqrt{29}/2\); the false option uses AB in place of BC.
A quick check: \(\sin^2 A + \cos^2 A = 25/29 + 4/29 = 1\), confirming sides 2, 5 and \(\sqrt{29}\). Since A and C are complementary, \(\cos C = \sin A\), which is why option C is true.
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