The value of tanθ (θ < 90°) increases as θ increases. Write ‘True’ or ‘False’ and justify your answer
Solution:
Given, θ < 90°
We have to determine if the value of tanθ increases as θ increases.
Consider a triangle ABC with perpendicular AC fixed.
So, ABC is a right triangle with C at right angle.
We know that tan A = opposite/adjacent
Here, AC = opposite
tanθ = AC/BC
As θ increases θ₁,
tanθ₁ = AC/B₁C
As θ increases to θ₂,
tanθ₂ = AC/B₂C
We observe that as θ increases to θ₁ and θ₂, BC decreases to B₁C and B₂C.
i.e.,as θ > θ₁ > θ₂ then BC < B₁C < B₂C
Therefore, the value of tanθ increases as θ increases.
✦ Try This: Prove that: sin(90° - θ) tanθ = sinθ
We have to prove that sin(90° - θ) tanθ = sinθ
By using trigonometric ratio of complementary angles,
sin(90° - A) = cos A
So, sin(90° - θ) = cos θ
Now, sin(90° - θ) tanθ = cos θ tan θ
We know that tan A = sin A/cos A
So, cos θ tan θ = cos θ (sin θ/cos θ)
= sin θ
Therefore, sin(90° - θ) tanθ = sinθ
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8
NCERT Exemplar Class 10 Maths Exercise 8.2 Sample Problem 2
The value of tanθ (θ < 90°) increases as θ increases. Write ‘True’ or ‘False’ and justify your answer
Summary:
The tangent function can be expressed as the ratio of the sine function and cosine function.The statement “The value of tanθ (θ < 90°) increases as θ increases” is true
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