Last updated on August 5th, 2025
In trigonometry, multiple angle formulas are used to express trigonometric functions of multiple angles in terms of functions of single angles. These formulas are crucial for simplifying expressions and solving equations. In this topic, we will learn the formulas for double, triple, and half angles.
The multiple angle formulas include double angle, triple angle, and half angle formulas.
Let’s learn the formulas to calculate these trigonometric expressions.
The double angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. They are given by:
- Sin(2θ) = 2sin(θ)cos(θ)
- Cos(2θ) = cos²(θ)
- sin²(θ) = 2cos²(θ)
- 1 = 1 - 2sin²(θ)
- Tan(2θ) = 2tan(θ) / (1 - tan²(θ))
The triple angle formulas express trigonometric functions of three times an angle in terms of functions of the original angle.
They are given by:
- Sin(3θ) = 3sin(θ) - 4sin³(θ) - Cos(3θ) = 4cos³(θ) - 3cos(θ) - Tan(3θ) = (3tan(θ) - tan³(θ)) / (1 - 3tan²(θ))
The half angle formulas express trigonometric functions of half an angle in terms of functions of the original angle. They are given by:
- Sin(θ/2) = ±√((1 - cos(θ))/2) - Cos(θ/2) = ±√((1 + cos(θ))/2) - Tan(θ/2) = ±√((1 - cos(θ))/(1 + cos(θ))) = sin(θ)/(1 + cos(θ)) = (1 - cos(θ))/sin(θ)
In math and real life, we use multiple angle formulas to simplify complex trigonometric expressions and solve equations. Here are some important uses:
- They allow us to solve trigonometric equations involving multiple angles.
- By learning these formulas, students can easily understand concepts related to calculus and higher-level mathematics.
- These formulas are essential in physics for wave and oscillation problems.
Students may find math formulas tricky and confusing. Here are some tips and tricks to master multiple angle formulas:
- Use mnemonic devices to remember formulas, such as “Double angle, double fun: 2sinθcosθ for the sine one.”
- Practice using these formulas in different problems to become familiar with their applications.
- Create flashcards with the formulas and practice them regularly for quick recall.
Students make errors when using multiple angle formulas. Here are some mistakes and ways to avoid them:
Calculate Sin(60°) using the double angle formula if Sin(30°) is known?
Sin(60°) = √3/2
Sin(60°) can be found using the double angle formula: Sin(2θ) = 2sin(θ)cos(θ). Here, θ = 30°: Sin(60°) = 2sin(30°)cos(30°) Sin(30°) = 1/2, and Cos(30°) = √3/2 So, Sin(60°) = 2(1/2)(√3/2) = √3/2
Find Cos(45°) using the half angle formula if Cos(90°) is known?
Cos(45°) = √2/2
Cos(45°) can be found using the half angle formula: Cos(θ/2) = ±√((1 + cos(θ))/2). Here, θ = 90°: Cos(45°) = ±√((1 + cos(90°))/2) Cos(90°) = 0 So, Cos(45°) = ±√((1 + 0)/2) = ±√(1/2) = √2/2
What is Tan(90°) if you use the double angle formula and Tan(45°) is known?
Tan(90°) is undefined.
Tan(90°) can be approached using the double angle formula: Tan(2θ) = 2tan(θ)/(1 - tan²(θ)). Here, θ = 45°: Tan(90°) = 2tan(45°)/(1 - tan²(45°)) Tan(45°) = 1 So, Tan(90°) = 2(1)/(1 - 1²) = 2/0, which is undefined.
Find the value of Sin(135°) using the double angle formula if Sin(67.5°) is known?
Sin(135°) = 1/√2
Sin(135°) can be found using the double angle formula: Sin(2θ) = 2sin(θ)cos(θ). Here, θ = 67.5°: Sin(135°) = 2sin(67.5°)cos(67.5°). Sin(67.5°) = cos(22.5°), and Cos(67.5°) = sin(22.5°) Substitute to get Sin(135°) = 1/√2
Calculate Cos(120°) using the triple angle formula if Cos(40°) is known?
Cos(120°) = -1/2
Cos(120°) can be found using the triple angle formula: Cos(3θ) = 4cos³(θ) - 3cos(θ). Here, θ = 40°: Cos(120°) = 4cos³(40°) - 3cos(40°). Cos(40°) = 1/2 So, Cos(120°) = 4(1/2)³ - 3(1/2) = 4(1/8) - 3/2 = -1/2
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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