Trigonometry (advanced) : More results for trigonometric values of Compound Angles

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More results for trigonometric values of Compound Angles

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Some Results

Quickly derive the identities. No need to memorize.

» Use $(a + b) (a - b)$ $(a+b)(a-b)$ $= a^{2} - b^{2}$ $=a^2-b^2$
      $\sin (A + B) \sin (A - B)$ $sin(A+B)sin(A-B)$
      $\cos (A + B) \cos (A - B)$ $cos(A+B)cos(A-B)$
      $\tan (A + B) \tan (A - B)$ $tan(A+B)tan(A-B)$
      $\cot (A + B) \cot (A - B)$ $cot(A+B)cot(A-B)$

» $3$ $3$ angles
$\sin (A + B + C)$ $sin(A+B+C)$ $= \sin (A + (B + C))$ $=sin(A+(B+C))$
$\cos (A + B + C)$ $cos(A+B+C)$ $= \cos (A + (B + C))$ $=cos(A+(B+C))$

result for sine

Consider $\sin (A + B) \sin (A - B)$ $sin(A+B)sin(A-B)$
$= (\sin A \cos B + \cos A \sin B)$ $quad = (sinA cosB + cosA sinB)$
$\times (\sin A \cos B - \cos A \sin B)$ $quad quad quad xx (sinA cosB - cosA sinB)$
$= \sin^{2} A \cos^{2} B - \cos^{2} A \sin^{2} B$ $quad = sin^2 A cos^2B - cos^2A sin^2B$

substitute $\cos^{2} B = (1 - \sin^{2} B)$ $cos^2 B = (1-sin^2 B)$
$= \sin^{2} A - \sin^{2} B \times (\sin^{2} A + \cos^{2} A)$ $quad = sin^2A - sin^2B xx(sin^2 A+cos^2 A)$
$= \sin^{2} A - \sin^{2} B$ $quad = sin^2A - sin^2B$

substitute $\sin^{2} A = (1 - \cos^{2} A)$ $sin^2A = (1-cos^2 A)$
$= \cos^{2} B - \cos^{2} A (\cos^{2} B + \sin^{2} B)$ $quad = cos^2B - cos^2A(cos^2B+sin^2B)$
$= \cos^{2} B - \cos^{2} A$ $quad = cos^2 B - cos^2 A$

$\sin (A + B) \sin (A - B)$ $sin(A+B)sin(A-B)$
$= \sin^{2} A - \sin^{2} B$ $quad quad = sin^2 A - sin^2 B$
$= \cos^{2} B - \cos^{2} A$ $quad quad = cos^2 B - cos^2 A$

result for cos

Consider $\cos (A + B) \cos (A - B)$ $cos(A+B)cos(A-B)$
$= (\cos A \cos B - \sin A \sin B)$ $quad = (cosA cosB - sinA sinB)$
$\times (\cos A \cos B + \sin A \sin B)$ $quad quad quad xx (cosA cosB + sinA sinB)$
$= \cos^{2} A \cos^{2} B - \sin^{2} A \sin^{2} B$ $quad = cos^2 A cos^2B - sin^2A sin^2B$

substitute $\cos^{2} B = (1 - \sin^{2} B)$ $cos^2 B = (1-sin^2 B)$
$= \cos^{2} A - \sin^{2} B \times (\sin^{2} A + \cos^{2} A)$ $quad = cos^2A - sin^2B xx(sin^2 A+cos^2 A)$
$= \cos^{2} A - \sin^{2} B$ $quad = cos^2A - sin^2B$

substitute $\sin^{2} B = (1 - \cos^{2} B)$ $sin^2B = (1-cos^2 B)$
$= \cos^{2} B (\cos^{2} A + \sin^{2} A) - \sin^{2} A$ $quad = cos^2B(cos^2A+sin^2A)-sin^2A$
$= \cos^{2} B - \sin^{2} A$ $quad = cos^2 B - sin^2 A$

$\cos (A + B) \cos (A - B)$ $cos(A+B)cos(A-B)$
$= \cos^{2} A - \sin^{2} B$ $quad quad = cos^2 A - sin^2 B$
$= \cos^{2} B - \sin^{2} A$ $quad quad = cos^2 B - sin^2 A$

result for tan

$\tan (A + B) \tan (A - B)$ $tan(A+B) tan(A-B)$
substitute
$\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ $tan(A+B) = (tan A + tan B)/(1- tan A tan B)$
$\tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ $tan(A-B) = (tan A - tan B)/(1+ tan A tan B)$

and multiply numerators and denominators of them
$= \frac{\tan^{2} A - \tan^{2} B}{1 - \tan^{2} A \tan^{2} B}$ $quad quad = (tan^2 A - tan^2 B)/(1-tan^2 A tan^2 B)$

$\tan (A + B) \tan (A - B)$ $tan(A+B) tan(A-B)$
$= \frac{\tan^{2} A - \tan^{2} B}{1 - \tan^{2} A \tan^{2} B}$ $quad quad = (tan^2 A - tan^2 B)/(1-tan^2 A tan^2 B)$

result for cot

$\cot (A + B) \cot (A - B)$ $cot(A+B) cot(A-B)$
Substitute
$\cot (A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B}$ $cot(A+B) = (cot A cot B -1)/(cot A + cot B)$

$\cot (A - B) = \frac{- \cot A \cot B - 1}{\cot A - \cot B}$ $cot(A-B) = (-cot A cot B -1)/(cot A - cot B)$

and multiply numerator and denominator
$= \frac{1 - \cot^{2} A \cot^{2} B}{\cot^{2} A - \cot^{2} B}$ $quad quad = (1-cot^2 A cot^2 B)/(cot^2 A - cot^2 B)$

$\cot (A + B) \cot (A - B)$ $cot(A+B) cot(A-B)$
$= \frac{1 - \cot^{2} A \cot^{2} B}{\cot^{2} A - \cot^{2} B}$ $quad quad = (1-cot^2 A cot^2 B)/(cot^2 A - cot^2 B)$

three angles

$\sin (A + B + C)$ $sin(A+B+C)$
$= \sin ((A + B) + C)$ $quad = sin( (A+B)+ C)$
$= \sin (A + B) \cos C + \cos (A + B) \sin C$ $quad = sin(A+B)cos C + cos(A+B)sinC$
$= \sin A \cos B \cos C + \cos A \sin B \cos C$ $quad = sinA cosB cos C + cosA sinB cos C$
$+ \cos A \cos B \sin C - \sin A \sin B \sin C$ $quad quad quad + cosA cosB sinC - sinA sinB sinC$

$\sin (A + B + C)$ $sin(A+B+C)$
$= \sin A \cos B \cos C + \cos A \sin B \cos C$ $quad = sinA cosB cos C + cosA sinB cos C$
$+ \cos A \cos B \sin C - \sin A \sin B \sin C$ $quad quad quad + cosA cosB sinC - sinA sinB sinC$

$\cos (A + B + C)$ $cos(A+B+C)$
$= \cos ((A + B) + C)$ $quad = cos( (A+B)+ C)$
$= \cos (A + B) \cos C - \sin (A + B) \sin C$ $quad = cos(A+B)cos C - sin(A+B)sinC$
$= \cos A \cos B \cos C - \sin A \sin B \cos C$ $quad = cosA cosB cos C - sinA sinB cos C$
$- \sin A \cos B \sin C - \cos A \sin B \sin C$ $quad quad quad - sinA cosB sinC - cosA sinB sinC$

$\cos (A + B + C)$ $cos(A+B+C)$
$= \cos A \cos B \cos C - \sin A \sin B \cos C$ $quad = cosA cosB cos C - sinA sinB cos C$
$- \sin A \cos B \sin C - \cos A \sin B \sin C$ $quad quad quad - sinA cosB sinC - cosA sinB sinC$

summary

More results of Trigonometric values for compound Angles $\sin (A + B) \sin (A - B)$ $sin(A+B)sin(A-B)$
$= \sin^{2} A - \sin^{2} B$ $quad quad = sin^2 A - sin^2 B$
$= \cos^{2} B - \cos^{2} A$ $quad quad = cos^2 B - cos^2 A$

$\cos (A + B) \cos (A - B)$ $cos(A+B)cos(A-B)$
$= \cos^{2} A - \sin^{2} B$ $quad quad = cos^2 A - sin^2 B$
$= \cos^{2} B - \sin^{2} A$ $quad quad = cos^2 B - sin^2 A$

$\tan (A + B) \tan (A - B)$ $tan(A+B) tan(A-B)$
$= \frac{\tan^{2} A - \tan^{2} B}{1 - \tan^{2} A \tan^{2} B}$ $quad quad = (tan^2 A - tan^2 B)/(1-tan^2 A tan^2 B)$

$\cot (A + B) \cot (A - B)$ $cot(A+B) cot(A-B)$
$= \frac{1 - \cot^{2} A \cot^{2} B}{\cot^{2} A - \cot^{2} B}$ $quad quad = (1-cot^2 A cot^2 B)/(cot^2 A - cot^2 B)$

$\sin (A + B + C) = \sin ((A + B) + C)$ $sin(A + B + C) = sin((A+B)+C)$

$\cos (A + B + C) = \cos ((A + B) + C)$ $cos(A + B + C) = cos((A+B)+C)$

Outline

It is advised to do the firmfunda version of "basics of Trigonometry" course before doing this.

The outline of material to learn "Advanced Trigonometry" is as follows.
Note: go to detailed outline of Advanced Trigonometry

→ Unit Circle form of Trigonmetric Values

→ Trigonometric Values in all Quadrants

→ Trigonometric Values or any Angles : First Principles

→ Understanding Trigonometric Values in First Quadrant

→ Trigonometric Values in First Quadrant

→ Trigonometric Values of Compound Angles: Geometrical Proof

→ Trigonometric Values of Compound Angles: Algebraic Proof

→ Trigonometric Values of Compound Angles: tan cot

→ Trigonometric Values of Compound Angles: more results