sin C C. cos A = cos C D. cos A = sin B
Select the true statement about triangle ABC. A. cos A = tan C B. cos A = sin C C. cos A = cos C D. cos A = sin B
The Correct Answer and Explanation is:
The correct answer is: D. cos A = sin B
Explanation:
In any triangle △ABC\triangle ABC, the relationship between angles and trigonometric functions is governed by the cofunction identities and the angle sum property of triangles. The sum of interior angles in any triangle is always: A+B+C=180∘A + B + C = 180^\circ
This means: A+B=180∘−CA + B = 180^\circ – C
From trigonometric cofunction identities, we know: sinθ=cos(90∘−θ)andcosθ=sin(90∘−θ)\sin \theta = \cos (90^\circ – \theta) \quad \text{and} \quad \cos \theta = \sin (90^\circ – \theta)
Let’s apply this identity to angles in triangle ABC. Since: A+B=180∘−CA + B = 180^\circ – C
Then, it follows that: B=180∘−C−AB = 180^\circ – C – A
Now consider: cosA=sin(90∘−A)\cos A = \sin (90^\circ – A)
But in a triangle, if angle B = 90° – A, then: cosA=sinB\cos A = \sin B
This is exactly what is given in option D: cos A = sin B, making it the true statement.
Why the other options are incorrect:
- A. cos A = tan C
This is not a standard trigonometric identity in triangles. Cosine and tangent relate differently and this equality is not generally true for arbitrary triangle angles. - B. cos A = sin C
Only true if A+C=90∘A + C = 90^\circ, which is not always the case in every triangle. - C. cos A = cos C
This would imply that A=CA = C, which is only true for an isosceles triangle where two angles are equal. This is not universally true.
Conclusion:
The only universally valid trigonometric identity among the options, applicable to any triangle, is cos A = sin B, due to cofunction identities and the triangle angle sum property.
