Respuesta :
Trigonometric ratios are sine, cosine, and tangent (opposite side over hypotenuse, adjacent side over hypotenuse, and opposite side over adjacent side, respectively); if you wanted to prove that one of the angles of the triangle is 90º, then the cosine of that angle would be 0, the sine would be 1, and the tangent would be undefined.
Answer with explanation:
The triangle in the Diagram Described has following measurement:
Longest Side = 65 units
One side which can be either Perpendicular or base = 63 units
And , other side which can be also, either Perpendicular or base = 16 units
We can prove that the triangle described is right triangle by two ways.
1. Using Converse of Pythagorean Theorem
Square of Longest side = Sum of Squares of other two sides-----(1)
So, Square of Longest Side = 65²=4225
Sum of Square of other two sides = 16² + 63²
= 256 + 3969
= 4225
Statement (1), is valid.
So,Triangle is right angled triangle, right angled at A.
2. using Trigonometric Ratios
Suppose the triangle is right Angled at A.
In Right triangle B AC
[tex]tan B=\frac{\text{Perpendicular}}{\text{Base}}\\\\tan B=\frac{16}{63}\\\\tan C=\frac{63}{16}\\\\ tan(B +C)=\frac{tan B + tan C}{1-tan B \times tanC}\\\\tan (B +C)=\frac{\frac{16}{63}+\frac{63}{16}}{1-\frac{16}{63}\times \frac{63}{16}}\\\\tan (B +C)=\frac{\text{Any rational number}}{0}\\\\tan (B +C)=\infty\\\\B +C=90^{\circ}\\\\ \text{Using the trigonometric Identity},tan(A+B)=\frac{tan A +tan B}{1-tan A*tan B}[/tex]
B +C =90°
Also,→ ∠A + ∠B + ∠C=180°≡ (Angle sum property of triangle)
→∠A +90°=180°
→∠A=180° -90°
→∠A=90°
So, triangle is right Angled triangle , Right angled at A.
Hence ,proved.
