cos θ = (a2+b2)/2ab, where a and b are two distinct numbers such that ab > 0. Write ‘True’ or ‘False’ and justify your answer
Solution:
Given, a and b are two distinct numbers such that ab > 0.
We have to determine if cos θ = (a² + b²)/2ab
Arithmetic mean > Geometric Mean
As AM = (a + b)/2
GM=√ab
So we get
(a² + b²)/2 > √a²b²
(a² + b²)/2 > ab
(a² + b²)/2ab > 1
It is given that
cos θ > 1
We know that the value of cos θ varies between 0 and 1.
Base/ Hypotenuse > 1
Base > Hypotenuse
As the value of cos cannot be greater than 1, the given statement is false.
Therefore, the given statement is false.
✦ Try This: If 3tan θ = 4 , show that (4 cos θ - sin θ)/(4 cos θ + sin θ) = 1/2.
Given, 3tan θ = 4
We have to prove that (4 cos θ - sin θ)/(4 cos θ + sin θ) = 1/2.
We know that tan A = sin A/cos A
tan θ = 4/3
sin θ = 4
cos θ = 3
So, (4 cos θ - sin θ) = 4(3) - 4 = 12 - 4 = 8
(4 cos θ + sin θ) = 4(3) + 4 = 12 + 4 = 16
Now, (4 cos θ - sin θ)/(4 cos θ + sin θ) = 8/16
= 1/2
Therefore, it is proved that (4 cos θ - sin θ)/(4 cos θ + sin θ) = 1/2.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8
NCERT Exemplar Class 10 Maths Exercise 8.2 Problem 10
cos θ = (a2+b2)/2ab, where a and b are two distinct numbers such that ab > 0. Write ‘True’ or ‘False’ and justify your answer
Summary:
The statement “cos θ = (a²+b²)/2ab, where a and b are two distinct numbers such that ab > 0” is false
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