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.

In this article, we will learn the Triple Angle Formulas for sine, cosine, tangent, cosecant, secant, and cotangent, their derivations, and applications.

What are Triple Angle Formulas?

Triple Angle Formulas, like their Double Angle counterparts, express trigonometric functions of three times an angle (3θ) in terms of functions of the original angle (θ). These formulas establish relationships between trigonometric functions involving triple angles and those related to single angles such as sine, cosine, tangent, cosecant, secant, and cotangent. Much like Double-Angle-Formulas, Triple-Angle-Formulas are derived from angle addition formulas and other trigonometric identities.

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Triple Angle Formulas in Trigonometry

In trigonometry, there are six trigonometric functions, hence accordingly there are three six triple angle formulas

• Sine Triple Angle Formula (Sin 3a)

• Cosine Triple Angle Formula (Cos 3a)

• Tangent Triple Angle Formula (Tan 3a)

• Cosecant Triple Angle Formula (Cosec 3a)

• Secant Triple Angle Formula (Sec 3a)

• Cotangent Triple Angle Formula (Cot 3a)

Sin 3a Formula

Sin 3a Formula is given as

sin(3a) = 3sin(a) – 4sin³(a)

Cos 3a Formula

Cos 3a formula is given as

cos(3a) = 4cos³(a) – 3cos(a)

Tan 3a Formula

Tan 3a formula is given as

tan(3a) = (3tan(a) – tan³(a)) / (1 – 3tan²(a))

Cosec 3a Formula

Cosec 3a formula is given as

cosec(3a) = 1 / (3sin(a) – 4sin³(a))

Sec 3a Formula

Sec 3a formula is given as

sec(3a) = 1 / (4cos³(a) – 3cos(a))

Cot 3a Formula

Cot 3a formula is given as

cot(3a) = (1 – 3tan²(a)) / (3tan(a) – tan³(a))

Triple Angle Formula 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))

Cosec(3θ) Proof

The proof of cosec 3θ is discussed below:

cosec(3θ) = 1 / sin(3θ) = 1 / (3sin(θ) – 4sin³(θ))

Sec(3θ) Proof

The proof of cosec 3θ is discussed below:

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

Cot(3θ) Proof

The proof of cosec 3θ is discussed below:

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

Triple Angle Identities

Triple angle identities in trigonometry is summarized below:

Trigonometric Function	Triple Angle Formula
sin(3θ)	3sin(θ) – 4sin3(θ)
cos(3θ)	4cos3(θ) – 3cos(θ)
tan(3θ)	(3tan(θ) – tan3(θ)) / (1 – 3tan2(θ))
csc(3θ)	1 / (3sin(θ) – 4sin3(θ))
sec(3θ)	1 / (4cos3(θ) – 3cos(θ))
cot(3θ)	(1 – 3tan2(θ)) / (3tan(θ) – tan3(θ))

Triple Angle Formula Conclusion

Triple Angle Formulas are crucial in trigonometry, establishing connections between trigonometric functions of three times an angle and those of the original angle. These formulas find extensive applications across various scientific, mathematical, and engineering disciplines, showcasing their importance in simplification, problem-solving, and analysis involving angles and periodic functions.

Triple Angle Formula Solved Examples

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

Solution:

sin(3θ θ) = 3sin(θ) – 4sin³(θ)

sin(3θ) = 3 × (1/2) – 4 × (1/2)³

sin(3θ) = 3/2 – 1/4

sin(3θ) = 5/4

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) ≈ 1.1064

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

Triple Angle Formula Practice question

Try out the following questions on triple angle formulas

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

Triple Angle Formulas FAQs

What are Triple Identities?

Triple Angle Identities are expansion of trigonometric functions when they are expressed in terms of thrice of an angle

What are Triple Angle Formulas for Sine and Cosine?

Triple angle formulas for sine and cosine are:

• sin(3θ) = 3sin(θ) – 4sin³(θ)
• cos(3θ)=4cos³(θ) – 3cos(θ)

What is the Formula of Tan 3a?

The formula for tan 3a is given as tan(3a) = (3tan(a) – tan³(a)) / (1 – 3tan²(a))

What is Sin 3x Formula?

Sin 3x formula is given as sin(3x) = 3sin(x) – 4sin³(x)

How to Prove Triple Angle Identities?

Triple Angle Identities can be proved using sum of Angle formula