Question

Which trigonometric ratios are correct for triangle ABC? Check all that apply

Answer 1

Consider an angle M with measure m≠90°, in a right triangle.

Let

OPP denote the length of the side opposite to M,
ADJ denote the length of the side adjacent to M, and
HYP denote the hypotenuse.

then:

Sin(M) = OPP/HYP
Cos(M)= ADJ/HYPP
Tan(M)=OPP/ADJ


Back to our problem,

using the Pythagorean we can find the length of AB:


`|AB|^2+|AC|^2=|BC|^2\\\\|AB|^2+9^2=18^2\\\\|AB|^2=18^2-9^2\\\\|AB|^2=(18-9)(18+9)=9 \cdot 27=9 \cdot 9 \cdot3\\\\|AB|=9 √(3)`


`Sin(C)= (OPP)/(HYP)=(9 √(3) )/(18)= ( √(3) )/(2)`

`Cos(B)= (ADJ)/(HYP)= (9 √(3))/(18)= ( √(3))/(2)`

`Tan(C)= (OPP)/(ADJ)= (9 √(3) )/(9)= √(3)`

`Sin(B)= (OPP)/(HYP)= (9)/(18)= (1)/(2)`

`Tan(B)= (OPP)/(ADJ)= (9)/(9 √(3))= (1)/( √(3) )= ( √(3))/(3)`


Answer: 1, 3, 4