Triple Angle Formulas

Triple Angle Formulas or Triple Angle Identities are an extension of the Double Angle Formulas in trigonometry. They express trigonometric functions of three times an angle in terms of functions of the original angle. Understanding these formulas is essential in solving complex trigonometric equations, simplifying expressions, and analyzing various mathematical and real-world problems.

Other Ratios

• Cosec(3θ): cosec(3θ) = 1 / sin(3θ) = 1 / (3sin(θ) - 4sin³(θ))
• Sec(3θ): sec(3θ) = 1 / cos(3θ) = 1 / (4cos³(θ) - 3cos(θ))
• Cot(3θ): cot(3θ) = 1 / tan(3θ) = (1 - 3tan²(θ)) / (3tan(θ) - tan³(θ))

Proof

The proof of triple angle formulas in trigonometry is mentioned below:

Sin(3θ) Proof

The proof of sin 3θ is discussed below:

sin(3θ) = sin(2θ + θ)

= sin(2θ)cos(θ) + cos(2θ)sin(θ)

= 2sin(θ)cos(θ)(1 - 2sin²(θ)) + (cos²(θ) - sin²(θ))sin(θ)

= 3sin(θ) - 4sin³(θ)

Cos(3θ) Proof

The proof of cos 3θ is discussed below:

Using the angle addition formula:

cos(3θ) = cos(2θ + θ)

= cos(2θ)cos(θ) - sin(2θ)sin(θ)

= (cos²(θ) - sin²(θ))cos(θ) - 2sin(θ)cos(θ)sin(θ)

= 4cos³(θ) - 3cos(θ)

Tan(3θ) Proof

The proof of tan 3θ is discussed below:

tan(3θ) = tan(2θ + θ)

= (tan(2θ) + tan(θ)) / (1 - tan(2θ)tan(θ))

= ((2tan(θ))/(1 - tan²(θ)) + tan(θ)) / (1 - (2tan(θ))/(1 - tan²(θ)) × tan(θ))

= (3tan(θ) - tan³(θ)) / (1 - 3tan²(θ))

Related Articles:

• Double Angle Formula
• Trigonometry Table
• Trigonometric Identities
• Trigonometry Formulas
• Trigonometry Ratios

Triple Angle Formula Solved Examples

Example 1: Find the value of sin(3θ) given sin(θ) = 1/2.

Solution:

sin(3θ) = 3sin(θ) - 4sin³(θ)

⇒ sin(3θ) = 3 × (1/2) - 4 × (1/2)³

⇒ sin(3θ) = 3/2 - 4/8

⇒ sin(3θ) = 3/2 - 1/2

⇒ sin(3θ) = 1

Example 2: Determine cos(3θ) if cos(θ) = -3/5.

Solution:

cos(3θ) = 4cos³(θ) - 3cos(θ)

⇒ cos(3θ) = 4 × (-3/5)³ - 3 × (-3/5)

⇒ cos(3θ) = 4 × (-27/125) + 9/5

⇒ cos(3θ) = -108/125 + 225/125

⇒ cos(3θ) = 117/125

Example 3: Calculate tan(3θ) given tan(θ) = 4.

Solution:

tan(3θ) = (3tan(θ) - tan³(θ)) / (1 - 3tan²(θ))

⇒ tan(3θ) = (3 × 4 - 4³) / (1 - 3 × 4²)

⇒ tan(3θ) = (12 - 64) / (1 - 48)

⇒ tan(3θ) = -52 / -47

⇒ tan(3θ) = 52/47

Example 4: If sec(θ) = -2, find sec(3θ)

Solution:

sec(3θ) = 1 / cos(3θ) = 1 / (4cos³(θ) - 3cos(θ))

⇒ sec(3θ) = 1 / (4 × (-2)³ - 3 × (-2))

⇒ sec(3θ) = 1 / (-32 + 6)

⇒ sec(3θ) = 1 / (-26)

⇒ sec(3θ) = -1/26

Example 5: Given cot(θ) = 7/24, determine cot(3θ)

Solution:

cot(3θ) = 1 / tan(3θ) = (1 - 3tan²(θ)) / (3tan(θ) - tan³(θ))

⇒ cot(3θ) = 1 / (3tan(θ) - tan³(θ))

⇒ cot(3θ) = 1 / (3 × 7/24 - (7/24)³)

⇒ cot(3θ) = 1 / (21/24 - 343/13824)

⇒ cot(3θ) = 1 / (5184/552)

⇒ cot(3θ) ≈ 0.1068

Practice Questions

Q1: If sin(θ) = 3/5, determine sin(3θ).

Q2: Given cos(θ) = -4/7, find cos(3θ).

Q3: Calculate tan(3θ) if tan(θ) = -1/3.

Q4: If csc(θ) = -13/5, what is the value of csc(3θ)?

Q5: Determine sec(3θ) if sec(θ) = 2.