In a right triangleABC right angled at B, AC = 3 root
5 and AB + BC = 9. Find all the six trigonometric ratios of angle c
Answers
Answer:
We have been given a right angled triangle in which B measures 90°, AC measures 3√5 and AB + BC is equal to 9.
Let's write given terms in the listed form.
- B = 90°
- AC = 3√5 cm
- AB + BC = 9 cm
To Find:
We have been asked to find all the six trigonometric ratios of angle C.
Method of Solution:
Let the sides AB and BC be measure x and y respectively.
- AB = x cm
- BC = y cm
- AC = 3√5 cm
- AB + BC = 9 cm ..... (i)
Step: Do use pythagoras theorem in order to make relation between the sides.
Pythagoras theorem is given by, (Hypotenuse)² = (Perpendicular)² + (Base)²
∴ AC² = AB² + BC²
Or, (3√5)² = AB² + BC²
Or, AB² + BC² = 45 ..... (ii)
Step: Use given terms to find the remaining sides of triangle.
AB² + BC² = 45
⇒(AB+BC)² - 2AB.BC = 45
⇒ (9)² - 2AB.BC = 45
⇒ 81 - 2AB.BC = 45
⇒ 2AB.BC = 36
⇒ AB.BC = 18 ..... (iii)
Since we have two equations, we will use them to find the value of unknown sides.
- AB + BC = 9 .... (i)
- AB.BC = 18 .... (iii)
After solving equations (i) and (iii), we get:
- AB = 6 cm
- BC = 3 cm
Step: Do consider the given right angled triangle in order to find the trigonometric ratio.
When we consider the right angled triangle in order to find the trigonometric ratio of angle C, we have the following outcomes.
- Perpendicular = 6 cm
- Base = 3 cm
- Hypotenuse = 3√5 cm
By definition, Sin C can be given by,
Sin C = Perpendicular/Hypotenuse
or, Sin C = 6/3√5
or, Sin C = 2/√5 cm
By definition, Cos C can be given by,
Cos C = Base/Hypotenuse
Or, Cos C = 3/2√5 cm
By definition, Tan C can be given by,
Tan C = Perpendicular/Base
or, Tan C = 6/3
or, Tan C = 2 cm
By definition, Cosec C can be given by,
Cosec C = Hypotenuse/Perpendicular
or, Cosec C = 3√5/6
or, Cosec C = √5/2 cm
By definition, Sec C can be given by,
Sec C = Hypotenuse/Base
or, Sec C = 2√5/3 cm
By definition, Cot C can be given by,
CotC = Base/Perpendicular
or, Cot C = 3/6
or, Cot C = 1/2 cm
