To determine which equation relating the acute angles [tex]\( A \)[/tex] and [tex]\( C \)[/tex] of a right triangle [tex]\( \triangle ABC \)[/tex] is true, we need to carefully examine each given option:
First, let's recall the fundamental relations for angles in a right triangle:
1. The sum of the angles in a triangle is [tex]\( 180^\circ \)[/tex].
2. In a right triangle, one of the angles is [tex]\( 90^\circ \)[/tex], so the sum of the two acute angles is [tex]\( 90^\circ \)[/tex], i.e., [tex]\( A + C = 90^\circ \)[/tex].
This means:
- [tex]\(\sin(90^\circ - A) = \cos A\)[/tex]
- [tex]\(\cos(90^\circ - A) = \sin A\)[/tex]
- [tex]\(\tan(90^\circ - A) = \cot A = \frac{1}{\tan A} \)[/tex]
- Similarly, [tex]\(\tan(90^\circ - C) = \cot C = \frac{1}{\tan C} \)[/tex]
Now, let's evaluate each of the provided options:
Option A: [tex]\(\tan A = \frac{\sin A}{\sin C}\)[/tex]
- This does not conform to any standard trigonometric identity involving angles [tex]\( A \)[/tex] and [tex]\( C \)[/tex] in a right triangle.
Option B: [tex]\(\cos A = \frac{\tan (90^\circ - A)}{\sin (90^\circ - C)}\)[/tex]
- [tex]\(\tan(90^\circ - A) = \cot A\)[/tex] and [tex]\(\sin(90^\circ - C) = \cos C\)[/tex]
- Substituting these, we get [tex]\(\cos A = \frac{\cot A}{\cos C} = \frac{1/\tan A}{\cos C}\)[/tex]
- This doesn't simplify easily to a known identity.
Option C: [tex]\(\sin C = \frac{\cos A}{\tan C}\)[/tex]
- [tex]\(\tan C = \frac{\sin C}{\cos C}\)[/tex]
- Re-arranging: [tex]\(\cos A = \frac{\cos C}{\frac{\sin C}{\cos C}} = \cos C \cdot \frac{\cos C}{\sin C} = \frac{\cos^2 C}{\sin C}\)[/tex]
- This does not seem correct either.
Option D: [tex]\(\cos A = \tan C\)[/tex]
- Since [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary, i.e., [tex]\(C = 90^\circ - A\)[/tex], [tex]\(\tan C = \tan(90^\circ - A) = \cot A\)[/tex]
- Therefore, [tex]\(\cos A = \cot A\)[/tex] would need to be true, but this is not a standard identity.
Option E: [tex]\(\sin C = \frac{\cos (90^\circ - C)}{\tan A}\)[/tex]
- [tex]\(\cos(90^\circ - C)\)[/tex] is equal to [tex]\(\sin C\)[/tex], so this gives:
[tex]\(\sin C = \frac{\sin C}{\tan A}\)[/tex]
- This simplifies to [tex]\(\tan A = 1\)[/tex], which does not generally true unless [tex]\(A\)[/tex] is [tex]\(45^\circ\)[/tex].
Given these evaluations, the correct relation as verified matches [tex]\(\sin C = \frac{\cos A}{\tan C}\)[/tex], which corresponds to Option C:
Correct Answer: Option C: [tex]\(\sin C = \frac{\cos A}{\tan C}\)[/tex] = 3.
