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.
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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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 is given as
sin(3a) = 3sin(a) – 4sin3(a)
Cos 3a formula is given as
cos(3a) = 4cos3(a) – 3cos(a)
Tan 3a formula is given as
tan(3a) = (3tan(a) – tan3(a)) / (1 – 3tan2(a))
Cosec 3a formula is given as
cosec(3a) = 1 / (3sin(a) – 4sin3(a))
Sec 3a formula is given as
sec(3a) = 1 / (4cos3(a) – 3cos(a))
Cot 3a formula is given as
cot(3a) = (1 – 3tan2(a)) / (3tan(a) – tan3(a))
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 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.
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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
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
What are Triple Identities?
Triple Angle Identities are expansion of trigonometric functions when they are expressed in terms of thrice of an angle
Triple angle formulas for sine and cosine are:
- sin(3θ) = 3sin(θ) – 4sin3(θ)
- cos(3θ)=4cos3(θ) – 3cos(θ)
The formula for tan 3a is given as tan(3a) = (3tan(a) – tan3(a)) / (1 – 3tan2(a))
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.
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.
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.
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.
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.
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