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

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 Tangent 3 Theta Formula, let’s first learn in brief about what is a trigonometric ratio.