Double Angle Formulas in Trigonometry
- In trigonometry, the double angle formulas are as follows:
- Double angle formula for sine$$ \sin 2a = 2 \sin a \cos a $$
- Double angle formula for cosine $$ \cos 2a = \cos^2 a - \sin^2 a = \begin{cases} 1 - 2 \sin^2 a \\ \\ 2 \cos^2(a)-1 \end{cases} $$
- Double angle formula for tangent$$ \tan 2a = \frac{2 \tan a}{1- \tan^2 a} $$
From the cosine double angle formula, we can derive two other useful formulas:
$$ \sin^2 a = \frac{1-\cos 2a}{2} $$
$$ \cos^2 a = \frac{1+\cos 2a}{2} $$
Let's now explore examples and proofs of these double angle formulas.
Double Angle Formula for Sine
The double angle formula for sine is $$ \sin 2a = 2 \sin a \cos a $$
This means that the sine of twice an angle is not simply twice the sine of the angle:
$$ \sin 2a \ne 2 \sin a $$
Example
The sine of 30° (or π/6 radians) is:
$$ \sin \frac{\pi}{6} = \frac{1}{2} $$
The sine of twice that angle is the sine of 60°:
$$ \sin (2 \cdot \frac{\pi}{6}) = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} $$
We can verify this by using the double angle formula for sine at 30°:
$$ \sin (2 \cdot \frac{\pi}{6}) = 2 \sin \frac{\pi}{6} \cos \frac{\pi}{6} = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} $$
Note: Doubling the sine of 30° yields a completely different result: $$ 2 \sin \frac{\pi}{6} = 2 \cdot \frac{1}{2} = 1 $$
Proof
The sine of twice an angle:
$$ \sin 2a $$
can be rewritten as:
$$ \sin 2a = \sin (a + a) $$
By applying the sine addition formula, we get:
$$ \sin 2a = \sin (a + a) = \sin a \cos a + \sin a \cos a $$
This simplifies to the formula we intended to prove:
$$ \sin 2a = 2 \sin a \cos a $$
Double Angle Formula for Cosine
The double angle formula for cosine is $$ \cos 2a = \cos^2 a - \sin^2 a = \begin{cases} 1 - 2 \sin^2 a \\ \\ 2 \cos^2(a)-1 \end{cases} $$
Thus, the cosine of twice an angle is not the same as twice the cosine of the angle:
$$ \cos 2a \ne 2 \cos a $$
Example
The cosine of 30° (or π/6 radians) is:
$$ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} $$
The cosine of twice that angle is the cosine of 60°:
$$ \cos (2 \cdot \frac{\pi}{6}) = \cos \frac{\pi}{3} = \frac{1}{2} $$
This can also be confirmed using the double angle formula for cosine at 30°:
$$ \cos (2 \cdot \frac{\pi}{6}) = \cos^2 \frac{\pi}{6} - \sin^2 \frac{\pi}{6} = \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2} $$
Note: Doubling the cosine of 30° gives a completely different result: $$ 2 \cos \frac{\pi}{6} = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} $$
Proof
The cosine of twice an angle:
$$ \cos 2a $$
can be rewritten as:
$$ \cos 2a = \cos (a + a) $$
By using the cosine addition formula, we find:
$$ \cos 2a = \cos (a + a) = \cos a \cos a - \sin a \sin a $$
This leads to the desired result:
$$ \cos 2a = \cos^2 a - \sin^2 a $$
From here, we can derive other related formulas. For instance, we can express cos2(a) as (1 - sin2(a)):
$$ \cos 2a = \cos^2 a - \sin^2 a $$
$$ \cos 2a = (1 - \sin^2 a) - \sin^2 a $$
$$ \cos 2a = 1 - 2\sin^2 a $$
Factoring out sin2(a), we get:
$$ \sin^2 a = \frac{1 - \cos 2a}{2} $$
Similarly, we can express sin2(a) as (1 - cos2(a)):
$$ \cos 2a = \cos^2 a - \sin^2 a $$
$$ \cos 2a = \cos^2 a - (1 - \cos^2 a) $$
$$ \cos 2a = -1 + 2 \cos^2 a $$
Factoring out cos2(a), we obtain:
$$ \cos^2 a = \frac{1 + \cos 2a}{2} $$
Double Angle Formula for Tangent
The double angle formula for tangent is $$ \tan 2a = \frac{2 \tan a}{1- \tan^2 a} $$
This shows that the tangent of twice an angle is not the same as twice the tangent of the angle:
$$ \tan 2a \ne 2 \tan a $$
Example
The tangent of 30° (or π/6 radians) is:
$$ \tan \frac{\pi}{6} = \frac{\sqrt{3}}{3} $$
The tangent of twice that angle is the tangent of 60°:
$$ \tan (2 \cdot \frac{\pi}{6}) = \tan \frac{\pi}{3} = \sqrt{3} $$
This can be confirmed using the double angle formula for tangent at 30°:
$$ \tan (2 \cdot \frac{\pi}{6}) = \frac{2 \cdot \tan \frac{\pi}{6}}{1 - \tan^2 \frac{\pi}{6}} = \frac{2 \cdot \frac{\sqrt{3}}{3}}{1 - \left(\frac{\sqrt{3}}{3}\right)^2} = $$ $$ = \frac{\frac{2 \sqrt{3}}{3}}{1 - \frac{3}{9}} = \frac{\frac{2 \sqrt{3}}{3}}{\frac{9-3}{9}} = \frac{\frac{2 \sqrt{3}}{3}}{\frac{2}{3}} = \frac{2 \sqrt{3}}{3} \cdot \frac{3}{2} = \sqrt{3} $$
Note: Doubling the tangent of 30° gives a different result: $$ 2 \tan \frac{\pi}{6} = 2 \cdot \frac{\sqrt{3}}{3} $$
Proof
The tangent of twice an angle:
$$ \tan 2a $$
can be rewritten as:
$$ \tan 2a = \tan (a + a) $$
By applying the tangent addition formula, we get:
$$ \tan 2a = \tan (a + a) = \frac{\tan a + \tan a}{1 - \tan a \tan a} $$
$$ \tan 2a = \frac{2\tan a}{1 - \tan^2 a} $$
And so on.
