Addition and subtraction formulas in trigonometry

In trigonometry, the formulas for adding and subtracting trigonometric functions (sine, cosine, tangent) are as follows:

Sine

$$ \sin (a+b) = \sin a \cos b + \cos a \sin b $$ $$ \sin (a-b) = \sin a \cos b - \cos a \sin b $$

Proof sine addition formula and sine subtraction formula

Cosine

$$ \cos (a+b) = \cos a \cos b - \sin a \sin b $$ $$ \cos (a-b) = \cos a \cos b + \sin a \sin b $$

Proof cosine addition formula and cosine subtraction formula

Tangent

$$ \tan (a+b) = \frac{\tan a + \tan b}{1- \tan a \tan b} $$ $$ \tan (a-b) = \frac{\tan a - \tan b}{1+ \tan a \tan b} $$

Proof tangent addition formula and tangent subtraction formula

    A practical example

    Let’s take two angles, a=30° and b=60°

    $$ \sin a = \sin 30° = \frac{1}{2} $$

    $$ \sin b = \sin 60° = \frac{\sqrt{3}}{2} $$

    The sine of the sum a+b is not simply the sum of the sines of the two angles:

    $$ \sin (a+b) \ne \sin(a) + \sin(b) $$

    $$ \sin (30°+60°) \ne \sin(30°) + \sin(60°) = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2} $$

    Note: The sine of 30°+60° is actually the sine of 90°. And as we all know, the sine of 90° is 1. So, it’s a common mistake to try and calculate the sine of the sum of two angles as sin(a+b) = sin(a) + sin(b).

    To correctly find the sine of the sum a+b, we use the sine addition formula:

    $$ \sin (a+b) = \sin a \cos b + \cos a \sin b $$

    Now substituting a=30° and b=60° gives us:

    $$ \sin (30°+60°) = \sin 30° \cos 60° + \cos 30° \sin 60° $$

    We already know that sin(30°) = 1/2 and sin(60°) = √3/2

    $$ \sin (30°+60°) = \frac{1}{2} \cdot \cos 60° + ( \cos 30° ) \cdot \frac{\sqrt{3}}{2} $$

    The cosine values for 30° and 60° are cos(30°) = √3/2 and cos(60°) = 1/2

    $$ \sin (30°+60°) = \frac{1}{2} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} $$

    $$ \sin (30°+60°) = \frac{1}{4} + \frac{3}{4} $$

    $$ \sin (30°+60°) = \frac{4}{4} $$

    $$ \sin (30°+60°) = 1 $$

    The sine of 30°+60° equals 1.

    The result is correct.

    And that’s how it works.

     
     

    Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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