The half-angle formula for tangent is: tan(A/2) = ±√[(1 - cos A) / (1 + cos A)] = (1 - cos A) / sin A = sin A / (1 + cos A).
Here's a breakdown of the formula and its variations:
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The Primary Formula:
tan(A/2) = ±√[(1 - cos A) / (1 + cos A)]
- This formula directly relates the tangent of half an angle (A/2) to the cosine of the full angle (A).
- The "±" sign indicates that you need to determine the correct sign based on the quadrant in which A/2 lies. This is because the tangent function can be positive or negative depending on the quadrant.
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Alternative Forms (often more convenient):
tan(A/2) = (1 - cos A) / sin A
- This form is often preferred because it avoids the square root, making calculations easier.
- It's derived from the primary formula using trigonometric identities.
tan(A/2) = sin A / (1 + cos A)
- This is another commonly used form that also avoids the square root.
- It's also derived from the primary formula using trigonometric identities.
Why are there different forms?
The different forms of the half-angle formula for tangent arise from algebraic manipulation and the use of other trigonometric identities. They are all equivalent but may be more convenient to use in different situations depending on what information you are given. The forms that avoid the square root are generally preferred for ease of calculation.
Example:
Let's say we want to find tan(π/8) using the half-angle formula. We know that π/8 is half of π/4. Therefore, A = π/4.
We know that sin(π/4) = √2/2 and cos(π/4) = √2/2.
Using the formula tan(A/2) = (1 - cos A) / sin A:
tan(π/8) = (1 - √2/2) / (√2/2) = (2 - √2) / √2 = (2√2 - 2) / 2 = √2 - 1.
In Summary:
The half-angle formula for tangent provides a way to calculate the tangent of half an angle if you know the sine and/or cosine of the full angle. It is a valuable tool in trigonometry and calculus.
