If sin A = 5/13. Find the value of 5tan²A + 4 CotA.
Answers
Given that,
- sin A = 5/13
We have to find the value of 5tan²A + 4 cotA
We know,
⇒ sin A = Perpendicular / Hypotenuse
⇒ 5/13 = Perpendicular / Hypotenuse
Therefore, comparing both sides, we have
- Hypotenuse = 13k
- Perpendicular = 5k
Imaging this in a right angled triangle.
I have attached the diagram.
Using the Pythagoras theorem in the triangle, we have
⇒ Hypotenuse² = Perpendicular² + Base²
⇒ (13k)² = (5k)² + Base²
⇒ 169k² - 25k² = Base²
⇒ 144k² = Base²
⇒ Base = 12k
Now, that we have perpendicular, base and hypotenuse, we can the required trigonometric ratios,
⇒ tan A = Perpendicular / Base
⇒ tan A = 5k / 12k
⇒ tan A = 5 / 12
Also,
⇒ cot A = 1 / tan A
⇒ cot A = 1 / (5/12)
⇒ cot A = 12 / 5
Now, Substituting the values of tan A and cot A in the given expression,
⇒ 5tan²A + 4cotA
⇒ 5(5/12)² + 4(12/5)
⇒ 125/144 + 48/5
⇒ (625 + 6912) / 720
⇒ 7537 / 720
Hence, The value of the given expression is 7537/720