Question

Show that the value of each of trigonometric ratio of an angle does not depends on a side of triangle it only depend on the angle

Answer 1

Answer:

Consider a triangle ABC in which angle B is 90 degree and angle A be α

Take a point P on AC and draw PQ perpendicular AB

Then, triangle APQ is similar to triangle ABC

Therefore, AQ/AB=AP/CP=PQ/CB

Then, PQ/AP = CB/AC=sin(α)

Similarly, AQ/AP= AB/AC=cos (α)

and, PQ/AQ= CB/AB= tan (α)

Similarly, if we produce AC to R and draw RS perpendicular AB produced then triangle ASR is similar to triangle ABC.

Therefore, AS/AB=AR/AC=RS/CB

Then, RS/AR=CB/AC=sin(α)

Similarly, AS/AR= AB/AC =cos (α)

and, RS/AS=CB/AB = tan (α)

Hence, the trignometric ratios of an angel do npt depends on the size of the triangle. They only depend on the angle