Tangent 3 Theta or tan 3 theta formula is tan 3θ = (3tanθ - tan3θ)/ (1 – 3tan2θ). It is an important trigonometric formula, that is used to solve various trigonometric problems. In this article we have covered, the Tangent 3 Theta (Tan 3θ) Formula, its derivation and others in detail.
Before, starting with the Tangent 3 Theta Formula, let's first learn in brief about what is a trigonometric ratio.
Trigonometric Ratios Definition
Trigonometric ratios are ratios of sides in a triangle and there are six trigonometric ratios. In a right-angle triangle, the six trigonometric ratios are defined as:
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Trigonometric Ratios- sin θ = (Opposite Side/Hypotenuse = AB/AC
- cos θ = Adjacent Side/Hypotenuse = BC/AC
- tan θ = Opposite side/adjacent side = AB/BC
- cosec θ = 1/sin θ = Hypotenuse/Opposite Side = AC/AB
- sec θ = 1/cos θ = Hypotenuse/Adjacent Side = AC/BC
- cot θ = 1/tan θ = Adjacent Side/Opposite Side = BC/AB
Tangent 3 Theta (Tan 3θ) Formula
Tan3θ is a triple angle identity in trigonometry. It is a crucial trigonometric identity that is used to solve a variety of trigonometric and integration issues. It is a trigonometric function that returns the tan function value for a triple angle. It may alternatively be written as tan3θ = sin 3θ/cos 3θ since the tangent function is a ratio of the sine and cosine functions.
Tangent 3 Theta FormulaThe value of tan 3θ repeats after every π/3 radians, tan 3θ = tan (3θ + π/3). Its graph is thinner than graph of tan θ and the graph of tan x and tan 3x is added below:
Tan 3θ Graphs\tan3\theta=\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}
Derivation on Tangent 3 Theta Formula
Formula for Tangent 3 theta is derived by using the sum angle formula for Tangent theta and Tangent 2 theta ratios.
To demonstrate that tan 3θ = (3 tan θ - tan3θ) / (1 - 3 tan2θ), we write 3θ as (2θ + θ).
L.H.S
= tan 3θ = tan (2θ + θ)
Use the formula tan (x + y) = (tan x + tan y) / (1 - tan x tan y)
= (tan 2θ + tan θ)/ (1 - tan 2θ tan θ)
Use the formula tan 2x = (2 tan x) / (1 - tan2x) for tan 2θ.
= [(2 tan θ / (1 - tan2θ)) + tan θ] / [1 - (2 tan θ / (1 - tan2θ)) tan θ]
= (tan θ - tan3θ + 2 tan θ) / (1 - tan2θ - 2 tan2θ)
= (3 tan θ - tan3θ) / (1 - 3 tan2θ)
= R.H.S.
This derives the formula for tangent 3 theta ratio.
Important Notes on Tan 3x Formula:
tan3x | (3 tan θ - tan3θ) / (1 - 3 tan2θ) |
|---|
d/dx (tan3x) | 3 sec2(3x) |
|---|
∫tan3x dx | (1/3) ln |sec 3x| + C |
|---|
Article Related to Tan3x Formula:
Problems on Tan 3x Formula
Problem 1. If tan θ = 3/4, find the value of tan 3θ using the formula.
Solution:
We have, tan θ = 3/4.
Using the formula we get,
tan 3θ = (3 tan θ - tan3θ) / (1 - 3 tan2θ)
= (3 (3/4) - (3/4)3) / (1 - 3 (3/4)2)
= (9/4 - 27/64) / (1 - 3 (9/16))
= (117/64) / (-11/16)
= -117/44
Problem 2. If tan θ = 12/5, find the value of tan 3θ using the formula.
Solution:
We have, tan θ = 12/5.
Using the formula we get,
tan 3θ = (3 tan θ - tan3θ) / (1 - 3 tan2θ)
= (3 (12/5) - (12/5)3) / (1 - 3 (12/5)2)
= (36/5 - 1728/125) / (1 - 3 (144/25))
= (-828/125) / (-407/25)
= 828/2035
Problem 3. If sin θ = 4/5, find the value of tan 3θ using the formula.
Solution:
We have, sin θ = 4/5.
Clearly cos θ = 3/5. Hence we have, tan θ = 4/3.
Using the formula we get,
tan 3θ = (3 tan θ - tan3θ) / (1 - 3 tan2θ)
= (3 (4/3) - (4/3)3) / (1 - 3 (4/3)2)
= (4 - 64/27) / (1 - 3 (16/9))
= (44/27) / (-13/3)
= -44/117
Problem 4. If cos θ = 12/13, find the value of tan 3θ using the formula.
Solution:
We have, cos θ = 12/13.
Clearly sin θ = 5/13. Hence we have, tan θ = 5/12.
Using the formula we get,
tan 3θ = (3 tan θ - tan3θ) / (1 - 3 tan2θ)
= (3 (5/12) - (5/12)3) / (1 - 3 (5/12)2)
= (5/4 - 125/1728) / (1 - 3 (25/144))
= (2035/1728) / (19/144)
= 2035/228
Problem 5. If sec θ = 17/8, find the value of tan 3θ using the formula.
Solution:
We have, sec θ = 17/8.
Find the value of tan θ using the formula sec2 θ = 1 + tan2 θ.
tan θ = √((289/64) - 1)
= √(225/64)
= 15/8
Using the formula we get,
tan 3θ = (3 tan θ - tan3θ) / (1 - 3 tan2θ)
= (3 (15/8) - (15/8)3) / (1 - 3 (15/8)2)
= (45/8 - 3375/1728) / (1 - 3 (225/64))
= (72/25) / (64/675)
= 243/8
Problem 6. Find the value of tan 135° using the tan 3x formula.
Solution:
We have to find the value of tan 135°.
Let us take 3x = 135
=> x = 135/3
=> x = 45°
We know, tan 45° = 1.
Using the tan 3x formula, we get
tan 135° = (3 tan 45° - tan345°) / (1- 3 tan245°)
= (3(1) - 13) / (1 - 3 (12))
= (3 - 1) / (1 - 3)
= 2 / (-2)
= -1
Problem 7. Find the value of tan 75° using the tan 3x formula.
Solution:
We have to find the value of tan 75°.
Let us take 3x = 75
=> x = 75/3
=> x = 25°
We know, tan 25° = 0.47.
Using the tan 3x formula, we get
tan 75° = (3 tan 25° - tan325°) / (1- 3 tan225°)
= (3(0.47) - (0.47)3) / (1 - 3 (0.47)2)
= (1.41 - 0.10) / (1 - 3 (0.22))
= (1.31) / (0.34)
= 3.85
Practice Problems on Tangent 3 Theta Formula
- Problem 1: Simplify the expression tan(3θ) if tan(θ)=1/√3.
- Problem 2: If tan(θ)=2, calculate tan(3θ) using the tangent triple angle formula.
- Problem 3: Prove that tan(3θ) is undefined when θ=π/6.
- Problem 4: Find θ if tan(3θ)=0 and 0≤θ<π/2.
- Problem 5: Verify the identity tan(3θ)=3tan(θ)−tan3(θ)/1−3tan2(θ) for θ=30∘.
- Problem 6: Solve for θ if tan(3θ)=tan(90∘−2θ).
- Problem 7: Prove that tan(3θ) can be expressed as tan(3θ)=3tan(θ)−tan3(θ)/1−3tan2(θ)) using the triple angle identities.
- Problem 8: If tan(θ)=1, what is tan(3θ)?
- Problem 9: Simplify the expression tan(3x)) given that tan(x)=−1/2
- Problem 10: Show that tan(3θ) is periodic with period π/3 by evaluating it at specific values of θ.
Conclusion
The tangent triple angle formula, tan(3θ)=3tan(θ)−tan3(θ) / 1−3tan2(θ), is a powerful identity in trigonometry that allows for the simplification of expressions and solution of equations involving tan(3θ). Understanding and applying this formula is crucial for solving complex trigonometric problems efficiently.
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