Table of Content

• What are Triple Angle Formulas?
• Triple Angle Formulas in Trigonometry
• Sin 3a Formula
• Cos 3a Formula
• Tan 3a Formula
• Cosec 3a Formula
• Sec 3a Formula
• Cot 3a Formula
• Triple Angle Formula Proof
• Sin(3θ) Proof
• Cos(3θ) Proof
• Tan(3θ) Proof
• Cosec(3θ) Proof
• Sec(3θ) Proof
• Cot(3θ) Proof
• Triple Angle Identities
• Triple Angle Formula Conclusion
• Triple Angle Formula Solved Examples
• Triple Angle Formula Practice question

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.

Also, Check

• Double Angle Formula
• Trigonometry Table
• Trigonometric Identities

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.

What are the Triple Angle Formulas in Trigonometry?

Triple angle formulas in trigonometry are equations that relate the sine, cosine, and tangent of three times an angle to the sine, cosine, and tangent of the original angle. These formulas are used to simplify calculations in various mathematical problems, especially in calculus and physics.

How is the Sine of a Triple Angle Expressed?

The sine of a triple angle, denoted as sin(3θ), can be expressed using the formula sin(3θ) = 3sin(θ) – 4sin³(θ). This formula helps in expanding trigonometric functions involving triple angles.

What is the Formula for Cosine of a Triple Angle?

The cosine of a triple angle, denoted as cos(3θ), is given by the formula cos(3θ) = 4cos³(θ) – 3cos(θ). This is useful for breaking down complex trigonometric expressions involving the cosine of triple angles.

Can the Tangent of a Triple Angle be Simplified Using a Formula?

Yes, the tangent of a triple angle can be simplified using the formula tan(3θ) = (3tan(θ) – tan³(θ)) / (1 – 3tan²(θ)). This formula allows for the simplification of expressions involving the tangent of triple angles and is particularly helpful in advanced algebra and calculus.

Why are Triple Angle Formulas Important in Mathematics?

Triple angle formulas are important because they provide a method to simplify complex trigonometric expressions and solve equations more efficiently. They are widely used in various fields of science and engineering, where trigonometric relationships play a crucial role in modeling and problem-solving.