Let's analyze the trigonometric ratios with respect to angle [tex]\(A\)[/tex] in a right triangle. Remember that the angles in a right triangle always add up to [tex]\(90^\circ\)[/tex], so the angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary ([tex]\(A + C = 90^\circ\)[/tex]).
1. Consider [tex]\(\cos A\)[/tex]:
- The cosine of angle [tex]\(A\)[/tex] ([tex]\(\cos A\)[/tex]) in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
- Clearly, [tex]\(\cos A\)[/tex] is not the same as [tex]\(\sin A\)[/tex], which is the ratio of the opposite side to the hypotenuse.
- Hence, [tex]\(\cos A \ne \sin A\)[/tex].
2. Consider [tex]\(\sin C\)[/tex]:
- Since angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary:
[tex]\[
C = 90^\circ - A
\][/tex]
- The sine of angle [tex]\(C\)[/tex] ([tex]\(\sin C\)[/tex]) is the same as the cosine of angle [tex]\(A\)[/tex] ([tex]\(\cos A\)[/tex]):
[tex]\[
\sin C = \cos A
\][/tex]
- Thus, [tex]\(\sin C\)[/tex] does not equal [tex]\(\sin A\)[/tex].
3. Consider [tex]\(\tan C\)[/tex]:
- The tangent of angle [tex]\(C\)[/tex] ([tex]\(\tan C\)[/tex]) is the ratio of the opposite side to the adjacent side for angle [tex]\(C\)[/tex].
- Since [tex]\(\tan C = \cot A\)[/tex] (the cotangent of [tex]\(A\)[/tex]):
[tex]\[
\tan C = \frac{1}{\tan A} = \frac{1}{\frac{\sin A}{\cos A}} = \frac{\cos A}{\sin A}
\][/tex]
- Clearly, [tex]\(\tan C\)[/tex] is not the same as [tex]\(\sin A\)[/tex].
4. Consider [tex]\(\cos C\)[/tex]:
- The cosine of angle [tex]\(C\)[/tex] ([tex]\(\cos C\)[/tex]) in the right triangle is:
[tex]\[
\cos C = \sin A
\][/tex]
- This is because for the complement of angle [tex]\(A\)[/tex] (which is [tex]\(C\)[/tex]):
[tex]\[
\cos C = \sin A
\][/tex]
- So, [tex]\(\cos C\)[/tex] is indeed equal to [tex]\(\sin A\)[/tex].
Based on the above analysis, the trigonometric ratio that will not have the same value as [tex]\(\sin A\)[/tex] is:
C. [tex]\(\tan C\)[/tex]
