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