BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon103 Learners

Last updated on August 5th, 2025

Math Whiteboard Illustration

Math Formulas for Multiple Angles

Professor Greenline Explaining Math Concepts

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.

Math Formulas for Multiple Angles for Canadian Students
Professor Greenline from BrightChamps

List of Math Formulas for Multiple Angles

The multiple angle formulas include double angle, triple angle, and half angle formulas.

 

Let’s learn the formulas to calculate these trigonometric expressions.

Professor Greenline from BrightChamps

Math Formula for Double Angles

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))

Professor Greenline from BrightChamps

Math Formula for Triple Angles

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))

Professor Greenline from BrightChamps

Math Formula for Half Angles

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)

Professor Greenline from BrightChamps

Importance of Multiple Angle Formulas

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.

Professor Greenline from BrightChamps

Tips and Tricks to Memorize Multiple Angle Formulas

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.

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them While Using Multiple Angle Formulas

Students make errors when using multiple angle formulas. Here are some mistakes and ways to avoid them:

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not Using the Correct Formula

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students sometimes apply the wrong formula for the given problem. To avoid this, ensure you understand which formula applies to double, triple, or half angles. Always double-check the problem context.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Sign Mistakes in Half Angle Formulas

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

The signs in half angle formulas can be confusing. Remember that the sign depends on the quadrant in which the angle lies. Be cautious and verify the angle's quadrant before applying the formula.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Forgetting to Square in Double Angle Formulas

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students sometimes forget to square the terms in the double angle formula for cosine. Remember that Cos(2θ) has three forms, and each involves squaring: cos²(θ) - sin²(θ), 2cos²(θ) - 1, or 1 - 2sin²(θ).

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing Double and Triple Angle Formulas

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often mix up double and triple angle formulas. To avoid this confusion, practice and understand the derivation and application of each formula.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Incorrect Simplification

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

In simplifying expressions using these formulas, students may make algebraic errors. Carefully work through each step and verify the calculations for accuracy.

arrow-right
Max from BrightChamps Saying "Hey"
Hey!

Examples of Problems Using Multiple Angle Formulas

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Calculate Sin(60°) using the double angle formula if Sin(30°) is known?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Sin(60°) = √3/2

Explanation

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

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 2

Find Cos(45°) using the half angle formula if Cos(90°) is known?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Cos(45°) = √2/2

Explanation

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

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 3

What is Tan(90°) if you use the double angle formula and Tan(45°) is known?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Tan(90°) is undefined.

Explanation

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.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 4

Find the value of Sin(135°) using the double angle formula if Sin(67.5°) is known?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Sin(135°) = 1/√2

Explanation

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

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 5

Calculate Cos(120°) using the triple angle formula if Cos(40°) is known?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Cos(120°) = -1/2

Explanation

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

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Ray Thinking Deeply About Math Problems

FAQs on Multiple Angle Math Formulas

1.What is the formula for double angles?

The double angle formulas are: - 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))

Math FAQ Answers Dropdown Arrow

2.What is the formula for triple angles?

The triple angle formulas are: - Sin(3θ) = 3sin(θ) - 4sin³(θ) - Cos(3θ) = 4cos³(θ) - 3cos(θ) - Tan(3θ) = (3tan(θ) - tan³(θ))/(1 - 3tan²(θ))

Math FAQ Answers Dropdown Arrow

3.How to find the half angle?

The half angle formulas are: - 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))

Math FAQ Answers Dropdown Arrow

4.What is the value of Sin(30°)?

The value of Sin(30°) is 1/2.

Math FAQ Answers Dropdown Arrow

5.What is the value of Tan(45°)?

The value of Tan(45°) is 1.

Math FAQ Answers Dropdown Arrow
Professor Greenline from BrightChamps

Glossary for Multiple Angle Formulas

  • Double Angle: A formula that expresses trigonometric functions of twice an angle in terms of functions of the original angle.

     
  • Triple Angle: A formula that expresses trigonometric functions of three times an angle in terms of functions of the original angle.

     
  • Half Angle: A formula that expresses trigonometric functions of half an angle in terms of functions of the original angle.

     
  • Sine: A trigonometric function that represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.

     
  • Cosine: A trigonometric function that represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
Math Teacher Background Image
Math Teacher Image

Jaskaran Singh Saluja

About the Author

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.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
UAE - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom