Parametric formulas in trigonometry
In trigonometry, parametric formulas allow us to compute the sine and cosine of an angle α based on the tangent of half the angle, α/2.
Parametric formula for sine: $$ \sin \alpha = \frac{2 \tan \frac{\alpha}{2} }{1+ \tan^2 \frac{\alpha}{2} } $$
Parametric formula for cosine: $$ \cos \alpha = \frac{1 - \tan^2 \frac{\alpha}{2} }{1+ \tan^2 \frac{\alpha}{2} } $$
If we introduce the parameter
$$ t = \tan \frac{ \alpha }{ 2 } $$
the formulas can be rewritten in a simpler parametric form:
$$ \sin \alpha = \frac{2 \cdot t }{1+ t^2 } $$ $$ \cos \alpha = \frac{1 - t^2 }{1+ t^2 } $$
The meaning remains unchanged.
Why use parametric formulas?
If you already know the value of the tangent of α/2, these formulas make it easy to find the sine and cosine of the angle α.
Practical example
Let's take the tangent of 45° (π/4 radians):
$$ \tan 45° = 1 $$
Using α/2=45°, we can calculate the sine of α=90° with the parametric sine formula:
$$ \sin \alpha = \frac{2 \cdot \tan \frac{\alpha}{2} }{1+ \tan^2 \frac{\alpha}{2} } $$
$$ \sin 90° = \frac{2 \cdot \tan 45° }{1+ \tan^2 45° } $$
Since tan 45° = 1
$$ \sin 90° = \frac{2 \cdot 1 }{1+ 1 } $$
$$ \sin 90° = \frac{2}{2} $$
$$ \sin 90° = 1 $$
This gives us the sine of 90° starting from the tangent of 45°.
Example 2
Starting again with the tangent of 45° (π/4 radians):
$$ \tan 45° = 1 $$
We can now calculate the cosine of α=90° using the parametric cosine formula with α/2=45°:
$$ \cos \alpha = \frac{1 - \tan^2 \frac{\alpha}{2} }{1+ \tan^2 \frac{\alpha}{2} } $$
$$ \cos 90° = \frac{1 - \tan^2 45° }{1+ \tan^2 45° } $$
Since tan 45° = 1
$$ \cos 90° = \frac{1 - 1 }{1+ 1 } $$
$$ \cos 90° = \frac{0}{2} $$
$$ \cos 90° = 0 $$
We’ve now calculated the cosine of 90° starting from the tangent of 45°.
The proof
Proof of the parametric formula for sine
The sine of an angle a
$$ \sin a $$
can be rewritten in terms of a/2 using the sine doubling formula:
$$ \sin a = 2 \cdot \sin \frac{a}{2} \cos \frac{a}{2} $$
The denominator is 1, which is usually omitted, but here it helps clarify the next steps in the proof.
$$ \sin a = \frac{ 2 \cdot \sin \frac{a}{2} \cos \frac{a}{2} }{1} $$
According to the fundamental trigonometric identity, the sum of the squares of sine and cosine equals 1, i.e., sin2(θ) + cos2(θ) = 1.
We can use this identity with the angle a/2 to replace the denominator:
$$ \sin a = \frac{ 2 \cdot \sin \frac{a}{2} \cos \frac{a}{2} }{ \sin^2 \frac{a}{2} + \cos^2 \frac{a}{2} } $$
This expression is equivalent to the previous one.
Now, applying the invariant property of fractions, we divide both the numerator and denominator by cos2(a/2):
$$ \sin a = \frac{ \frac{ 2 \cdot \sin \frac{a}{2} \cos \frac{a}{2} }{ \cos^2 \frac{a}{2} } }{ \frac { \sin^2 \frac{a}{2} + \cos^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } } $$
Simplifying the cosine in the numerator gives us:
$$ \sin a = \frac{ \frac{ 2 \cdot \sin \frac{a}{2} }{ \cos \frac{a}{2} } }{ \frac { \sin^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } + 1 } $$
We further simplify the denominator with some algebraic steps:
$$ \sin a = \frac{ \frac{ 2 \cdot \sin \frac{a}{2} }{ \cos \frac{a}{2} } }{ \frac { \sin^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } + \frac { \cos^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } } $$
$$ \sin a = \frac{ \frac{ 2 \cdot \sin \frac{a}{2} }{ \cos \frac{a}{2} } }{ \frac { \sin^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } + 1 } $$
Since the sine-to-cosine ratio is the tangent, we arrive at:
$$ \sin a = \frac{ 2 \cdot \tan \frac{a}{2} }{ \tan^2 \frac{a}{2} + 1 } $$
Thus, we’ve derived the parametric sine formula.
Proof of the parametric formula for cosine
The cosine of an angle a
$$ \cos a $$
can also be rewritten in terms of a/2 using the cosine doubling formula:
$$ \cos a = \cos^2 \frac{a}{2} - \sin^2 \frac{a}{2} $$
Again, we explicitly include the denominator (1) for clarity:
$$ \cos a = \frac{ \cos^2 \frac{a}{2} - \sin^2 \frac{a}{2} }{1} $$
Using the fundamental trigonometric identity, where sin2 + cos 2 = 1, we rewrite the denominator as:
$$ \cos a = \frac{ \cos^2 \frac{a}{2} - \sin^2 \frac{a}{2} }{ \sin^2 \frac{a}{2} + \cos^2 \frac{a}{2} } $$
Next, we divide both the numerator and denominator by cos2(a/2):
$$ \cos a = \frac{ \frac{ \cos^2 \frac{a}{2} - \sin^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } }{ \frac{ \sin^2 \frac{a}{2} + \cos^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } } $$
Simplifying both the numerator and denominator yields:
$$ \cos a = \frac{ \frac{ \cos^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } - \frac{ \sin^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } }{ \frac{ \sin^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } + \frac{ \cos^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } } $$
$$ \cos a = \frac{ 1 - \frac{ \sin^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } }{ \frac{ \sin^2 \frac{a}{2} }{ \cos^2 \frac{a}{2} } + 1 } $$
Since the sine-to-cosine ratio is the tangent, we derive:
$$ \cos a = \frac{ 1 - \tan^2 \frac{a}{2} }{ \tan^2 \frac{a}{2} + 1 } $$
Thus, we’ve derived the parametric cosine formula.
And so on.