Identities

Identities#

There is a set of trigonometry identities. This page discusses proofs for some of them.

Sum & Difference#

Sum#

Validity of the identieties:

  • \(\sin(\alpha + \beta) = \sin{\alpha} \cos {\beta} + \cos{\alpha} \sin{\beta}\).

  • \(\cos(\alpha + \beta) = \cos{\alpha}\cos{\beta} - \sin{\alpha}\sin{\beta}\).

Could be illustrated using following drawing:

F B E C D A β α α α+β 1 cos β sin β sin(α+β) cos(α+β) sin α sin β cos α sin β sin α cos β cos α cos β

Step-by-step breakdown of this drawing:

  1. Draw a right triangle \(\Delta ABC\) with an angle of \(\angle BAC = \beta\) and a hypotenuse of length 1. The cathetes of this triangle are equal to \(\sin(\beta)\) and \(\cos(\alpha)\).

  2. Draw rectangle \(AFED\) around a triangle \(ABC\) so that \(\angle CAD = \alpha\).

  3. Using the properties of parallel lines:

    • \(\angle FBA = \angle BAC + \angle CAD = \beta + \alpha\).

    • \(\angle CAD = \angle BCE = \alpha\).

  4. Consider the triangle \(\Delta AFB\). Using the definitions of sine and cosine:

    • \(AF=\sin{(\alpha + \beta)}\).

    • \(FB=\cos{(\alpha + \beta)}\).

  5. Consider the triangle \(\Delta BEC\). Using the definitions of sine and cosine:

    • \(BE = \sin{\alpha} \sin{\beta}\).

    • \(CE = \cos{\alpha} \sin{\beta}\).

  6. Consider the triangle \(\Delta ACD\). Using the definitions of sine and cosine:

    • \(CD = \sin{\alpha} \cos{\beta}\).

    • \(AD = \cos{\alpha} \cos{\beta}\).

Finally, using the properties of rectangle:

  • \(AF = ED \Rightarrow \sin{(\alpha + \beta)} = \cos{\alpha} \sin{\beta} + \sin{\alpha} \cos{\beta}\).

  • \(FE = AD \Rightarrow \cos{(\alpha + \beta)} + \sin{\alpha}\cos{\beta} = \cos{\alpha}\cos{\beta} \Rightarrow \cos{(\alpha + \beta)} = \cos{\alpha}\sin{\beta} - \sin{\alpha}\cos{\beta}\).

Technically, speaking this figure does not prove identities - it works only for \(\alpha, \beta\) that \(\alpha + \beta < \pi/2, \alpha < pi/2, \beta < \pi/2\). However, it can be a useful way to remember the identities.

Difference#

Validity of identities:

  • \(\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\cos(\beta)\).

  • \(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\cos(\beta)\).

Could be illustrated using following drawing:

A B C E D F α β α - β α α 1 cos β cos α cos β sin α sin β sin β sin α cos β cos α sin β sin (α - β) cos (α - β)

There is a right triangle \(AEC\) with \(\angle AEC = \frac{\pi}{2}\) and hypotenuse \(AC=1\), inscribed in the triangle \(AFDB\) such that \(E\) lies on \(FD\), \(C\) lies on \(BD\) and \(\angle EAC = \alpha\).

To see correctness of sin/cos difference formulas follow the logic:

  • Due to the definitions of sine and cosine: \(AE = \sin{\beta}\) and \(EC = \cos{\beta}\).

  • Using the properties of parallel lines \(AB\) and \(FD\): \(\angle AEF = \angle BEA \Rightarrow \angle AEF = \alpha\).

  • Due to the properties of the right angle triangles and the definition of trigonometric functions:

    • For triangle \(AFE\): \(AF = \sin{\alpha}\cos{\beta}\), \(EF = \cos{\alpha}\cos{\beta}\).

    • For triangle \(EDC\): \(ED = \sin{\alpha}\sin{\beta}\), \(CD = \sin{\alpha}cos{\beta}\).

  • Since \(\angle ABC = \angle EAB - \angle EAC\): \(\angle EAC = \alpha - \beta\). Finally:

    • For triangle \(ABC\): \(BC=\sin{(\alpha - \beta)}\), \(AB=\cos{(\alpha - \beta)}\).

Finally, using properties of the rectangle:

  • \(AB = FE + ED \Rightarrow \cos{(\alpha - \beta)} = \cos{\alpha} \cos{\beta} + \sin{\alpha} \sin{\beta}\)

  • \(AF = BC + CD \Rightarrow \sin{\alpha}\cos{\beta} = \sin{(\alpha - \beta)} + \cos{\alpha} \sin{\beta} \Rightarrow \sin{(\alpha - \beta)} = \sin{\alpha}\cos{\beta} - \cos{\alpha}\sin{\beta}\).

This is not general proof - this picture only applies for \(0 \leq \alpha \leq \frac{\pi}{2}\) and \(\beta < \alpha\). How ever it’s a good way to remember the idea.