Half Angle Formulas

Half angle formulas in trigonometry are extremely useful tools that allow us to find the trigonometric values of half angles using known values of full angles. These formulas are derived from the double angle identities and are central to solving a variety of trigonometric problems, especially those that involve integration or solving trigonometric equations.

Understanding the half angle formulas requires familiarity with basic trigonometric identities, such as the Pythagorean identities, the double angle formula, and the fundamental trigonometric functions: sine, cosine, and tangent.

Basic trigonometric identities

Before diving into the half angle formulas, let's briefly summarize some basic identities:

• Pythagorean Identity:

  sin2(θ) + cos2(θ) = 1

• Double angle formula:

  sin(2θ) = 2sin(θ)cos(θ) cos(2θ) = cos2(θ) - sin2(θ) = 2cos2(θ) - 1 = 1 - 2sin2(θ)

• Basic trigonometric functions:

  sin(θ), cos(θ), tan(θ) = sin(θ)/cos(θ)

Half angle formula

Half angle formulas can be derived from double angle formulas. The idea is to express one angle as half of another angle. The half angle formulas for sine, cosine, and tangent are given as:

• Sine Half Angle Formula:

  sin(θ/2) = ±√((1 - cos(θ))/2)

• Cosine Half Angle Formula:

  cos(θ/2) = ±√((1 + cos(θ))/2)

• Tangent Half Angle Formula:

  tan(θ/2) = ±√((1 - cos(θ))/(1 + cos(θ))) or tan(θ/2) = sin(θ)/(1 + cos(θ)) or tan(θ/2) = (1 - cos(θ))/sin(θ)

The ± sign in these formulas is determined depending on the quadrant in which the resulting angle lies. Remember that every trigonometric function is positive or negative depending on the quadrant:

• Quadrant I: sin, cos, tan are positive
• Quadrant II: Sin is Positive
• Quadrant III: tan is positive
• Fourth quadrant: cos is positive

Visual example

To better understand the half angle concept, let's consider the unit circle. Imagine an angle θ on the unit circle. The angle θ/2 will lie between the origin (0 degrees) and the angle θ. The half angle formula lets us predict the sine, cosine, and tangent of this θ/2 angle based on the coordinates or angle that can be visually represented on the unit circle.

In the above diagram, let's say θ is 60 degrees, so half of it θ/2 is equal to 30 degrees. We can see this visually and calculate the sin, cos, and tan values for these using the half-angle formulas.

For example, calculate the sine of 30 degrees using the cosine of 60 degrees:

cos(60) = 1/2 sin(30) = √((1 - 1/2)/2) = √(1/4) = 1/2

Examples of half angle formulas in practice

Let's do some exercises to get a clear idea of how half angle identities work:

Example 1: Find sin(75°)

The angle 75° is not a standard angle. However, we can express it as follows:

75° = 150° / 2

Using the cosine of 150°, which is -√3/2, we calculate:

cos(150°) = -√3/2 sin(75°) = √((1 + cos(150°))/2) = √((1 - √3/2)/2) = √((2 - √3)/4)

This expression gives the value of sin(75°) directly without the need for further conversion.

Example 2: Find tan(22.5°)

We use the fact that 22.5° is half of 45°. Let's use the formula:

tan(θ/2) = (1 - cos(θ)) / sin(θ)

Here, θ = 45° and cos(45°) = √2/2, and sin(45°) = √2/2.

tan(22.5°) = (1 - √2/2) / (√2/2) = (2 - √2)/√2 = √2 (2 - √2) / 2 = (2√2 - 2) / 2

How to choose the right sign

As mentioned earlier, the sign (±) in the half-angle formulas depends on the quadrant where θ/2 is located. Consider the properties of the original quadrant of the angle and the unit circle. Here's how you can decide:

• Quadrant I: If θ lies in quadrant II, then θ/2 lies in quadrant I, and all trigonometric functions are positive.
• Quadrant II: If θ lies in quadrant III, then θ/2 will fall in quadrant II, where sine is positive, cosine and tangent are negative.
• Quadrant III: If θ lies in quadrant IV, then θ/2 will fall in quadrant III, where tangent is positive, sine and cosine are negative.
• Fourth quadrant: If θ is in quadrant I, then θ/2 will fall in quadrant IV, where cosine is positive, sine and tangent are negative.

Always visualize where θ/2 is on the unit circle or use known angle values to determine the proper sign.

Conclusion

Half angle formulas are important for simplifying trigonometric expressions and solving integrals. Becoming proficient in their use and understanding when to apply which sign is important in solving trigonometric problems. To gain fluency in using half angle identities, practice with different angles and try to obtain these angles found in various trigonometric functions.

Through repeated practice and applications, these formulas will become an invaluable part of your trigonometry toolkit.

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