Half Angle Calculator

The half angle calculator helps you find the exact values of all trigonometric identities of the half-angle such as sin, cos, and tan.

Enter the θ angle in degrees and press the button to convert it into radian. You will be able to see the Sin(θ/2), Cos(θ/2), and Tan(θ/2) of the half angle.

You can also click the show steps button to see how these values were calculated.

• Half angle formula of sin: sin A/2 = ±√[(1 – cos A) / 2]
• Half angle formula of cos: cos A/2 = ±√[(1 + cos A) / 2]
• Half angle formula of tan: tan A/2 = ±√[1 – cos A] / 1 + cos A sin A / (1 + cos A) (or) (1 – cos A) / sin A.

You might be interested in finding the octagon vertices or polar coordinates.

What is a Half Angle Calculator?

A half angle calculator is an online tool that allows users to calculate the sine, cosine, and tangent values of half an angle. This calculator utilizes half angle identities, which are fundamental trigonometric formulas that relate trigonometric functions of an angle to functions of half that angle. These identities are particularly useful in various mathematical and real-world applications.

The Importance of Half Angle Identities in Trigonometry

Half angle identities are essential in trigonometry for several reasons:

1. Simplification: They help simplify complex trigonometric expressions.
2. Problem-solving: These identities are crucial in solving trigonometric equations and proofs.
3. Applications: Half angle formulas have practical applications in fields such as physics, engineering, and computer graphics.

How the Half Angle Calculator Works

The half angle calculator operates based on the following half angle formulas:

1. Sin(θ/2) = ±√[(1 – cos θ) / 2]
2. Cos(θ/2) = ±√[(1 + cos θ) / 2]
3. Tan(θ/2) = ±√[(1 – cos θ) / (1 + cos θ)]

To use the calculator, follow these steps:

1. Enter the value of the full angle (θ) in degrees.
2. Click the “Convert to Radian” button.
3. The calculator will display the results for sin(θ/2), cos(θ/2), and tan(θ/2).

Understanding Half Angle Formulas

The Sin(θ/2) Formula

The sine half angle formula is derived from the cosine double angle formula. It allows us to express the sine of half an angle in terms of the cosine of the full angle:

Sin(θ/2) = ±√[(1 – cos θ) / 2]

This formula is particularly useful when we need to find the sine value of half an angle but only have information about the full angle.

The Cos(θ/2) Formula

Similar to the sine half angle formula, the cosine half angle formula is derived from the cosine double angle formula:

Cos(θ/2) = ±√[(1 + cos θ) / 2]

This formula allows us to calculate the cosine of half an angle when we know the cosine of the full angle.

The Tan(θ/2) Formula

The tangent half angle formula is derived from the sine and cosine half angle formulas:

Tan(θ/2) = ±√[(1 – cos θ) / (1 + cos θ)]

This formula is useful when we need to find the tangent of half an angle given information about the full angle.

Practical Applications of Half Angle Formulas

Half angle formulas have numerous practical applications in various fields:

1. Physics: These formulas are used in optics to calculate angles of reflection and refraction.
2. Engineering: Half angle formulas are applied in structural analysis and design.
3. Computer Graphics: They play a role in 3D modeling and animation.
4. Surveying: Half angle calculations are used in land surveying and navigation.

Example 1: Using the Half Angle Calculator

Let’s work through an example to illustrate how to use the half angle calculator:

Suppose we want to find the sine, cosine, and tangent of half of a 60° angle.

1. Enter 60 in the “Enter θ angle in degree” field.
2. Click the “Convert to Radian” button.
3. The calculator will display the following results: Sin(30°) ≈ 0.5
   Cos(30°) ≈ 0.866
   Tan(30°) ≈ 0.577

These values are approximate and rounded for simplicity. The calculator provides more precise results.

Example 2: Solving a Trigonometric Problem

Let’s consider a more complex problem that demonstrates the utility of half angle formulas:

Problem: In a right-angled triangle, one of the angles is 60°. Find the value of sin(30°) without using a calculator.

Solution:

1. We know that 30° is half of 60°.
2. We can use the half angle formula for sine:
   Sin(θ/2) = ±√[(1 – cos θ) / 2]
3. We need to know cos(60°). In a 30-60-90 triangle, cos(60°) = 1/2.
4. Substituting this into our formula:
   Sin(30°) = ±√[(1 – 1/2) / 2]
   = ±√(1/4)
   = ±(1/2)
5. Since we’re dealing with the first quadrant, we take the positive value.

Therefore, sin(30°) = 1/2, which matches our calculator results.

References

• Double‐Angle and Half‐Angle Identities. (n.d.). Double‐Angle and Half‐Angle Identities. cliffsnotes.com/study-guides/trigonometry/trigonometric-identities/double-angle-and-half-angle-identities
• www2.clarku.edu/faculty/djoyce/trig/identities.html. (n.d.). www2.clarku.edu/faculty/djoyce/trig/identities.html