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Answers
Answer:
tan60°= √3 is the answer for the given problem
Question
Using the formula, tan 2A = (2tan A)/(1-tan² A), find the value of tan 60°, it begin given that tan 30° = 1/√3
ANSWER
Given : -
Using the formula, tan 2A = (2tan A)/(1-tan² A) .
Required to find : -
- value of tan 60° ?
Condition mentioned : -
Use the formula;
tan 2A = (2tan A)/(1-tan² A)
Solution : -
tan 2A = (2tan A)/(1-tan² A)
We need to find the value of tan 60°
So,
A = 30°
This implies;
➙ tan 2(30°) = (2tan 30°)/(1-tan² 30°)
➙ tan 60° = (2tan 30°)/(1-tan² 30°)
We know that;
- tan 30° = 1/√3
➙ tan 60° = (2 x 1/√3 )/(1-[1/√3]² )
➙ tan 60° = (2/√3)/(1-1/3)
➙ tan 60° = (2/√3)/([3-1]/[3])
➙ tan 60° = (2/√3)/(2/3)
➙ tan 60° = (2/√3)÷(2/3)
➙ tan 60° = (2/√3)x(3/2)
➙ tan 60° = 2/√3 x 3/2
➙ tan 60° = 3/√3
Here,
Let's rationalize the denominator !
➙ Rationalising factor of √3 = √3
Multiply the numerator and denominator with rationalising factor
➙ tan 60° = 3/√3 x √3/√3
➙ tan 60° = (3√3)/(√3)²
➙ tan 60° = (3√3)/(3)
➙ tan 60° = √3
Therefore,
➙ Value of tan 60° = √3
Additional Information
We can also find the value of tan 60° using the formula;
tan (A+B) = (tan A+tan B)/(1-tan A tan B
Let's find out how we can find the value of tan 60° using this formula .
Consider the formula;
tan (A+B) = (tan A+tan B)/(1-tanA tan B)
Here,
A = 30°
B = 30°
➙ tan (30°+30°) = (tan 30°+tan 30°)/(1-tan 30° tan 30°)
➙ tan 60° = (tan 30°+tan 30°)/(1-tan 30° tan 30°)
We know that;
- tan 30° = 1/√3
➙ tan 60° = (1/√3+1/√3)/(1-1/√3 x 1/√3)
➙ tan 60° = ([1+1]/[√3])/(1 - 1/3 )
➙ tan 60° = (2/√3)/([3-1]/[3])
➙ tan 60° = (2/√3)/(2/3)
➙ tan 60° = (2/√3)÷(2/3)
➙ tan 60° = (2/√3) x (3/2)
➙ tan 60° = 3/√3
Now,
Let's rationalize the denominator
R.F. of √3 = √3
This implies;
➙ tan 60° = 3/√3 x √3/√3
➙ tan 60° = (3√3)/([√3]²)
➙ tan 60° = (3√3)/(3)
➙ tan 60° = √3
Hence,
➙ Value of tan 60° = √3