Question

If C and Z are acute angles and that cos C = cos Z prove that ∠C = ∠Z

Answer 1

Hello mate! Here is your answer.

Kindly please refer to the attachment.

I have considered a triangle CAZ where,

• ∠A = Right angle

• ∠C and ∠Z = Acute angle

By applying the method of trigonometric ratios and considering the given things:

$\implies{\rm}$ cos C = cos Z

$\implies{\rm}$ AC/CZ = AZ/CZ

$\implies{\rm}$ AC = AZ

As we can see that the opposite and adjacen side to 90° are equal. This says that the triangle assumed is a right-angled isosceles triangle.

In a right-angled isosceles triangle, we know that two angles are always equal (other than 90°). Therefore, we can conclude that:

$\implies{\rm}$ ∠C = ∠Z

Note: You can also use two triangles instead of one triangle and use similarity criterion method. According to me, this method looks simple!