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Half/Double/Multiple angle formula

Half/Double/Multiple angle formula

Half angle formula:

$$ 1.\ \sin \frac A2 = \pm \sqrt \frac{(1- \cos A)}{2} $$
$$ 2.\ \cos \frac A2 = \pm \sqrt \frac{(1+ \cos A)}{2} $$
$$ 3.\ \tan \frac A2 = \pm \sqrt \frac{(1- \cos A)}{(1+ \cos A)}= \frac{(1- \cos A)}{\sin A} = \frac {\sin A}{(1+ \cos A)} = \text{cosec} \ A – \cot A $$
$$ 4.\ \cot \frac A2 = \pm \sqrt \frac{(1+ \cos A)}{(1- \cos A)}= \frac{(1+ \cos A)}{\sin A} = \frac {\sin A}{(1- \cos A)} = \text{cosec} \ A + \cot A $$
5. sinA=2tanA21+tan2A2 5.\ \sin A= \frac {2 \tan \frac A2}{1+ \tan^2 \frac A2}
6. cosA=1–tan2A21+tan2A2 6.\ \cos A= \frac {1- \tan^2 \frac A2}{1+ \tan^2 \frac A2}
7. tanA=2tanA21–tan2A2 7.\ \tan A= \frac {2 \tan \frac A2}{1- \tan^2 \frac A2}
8. cotA=1–tan2A22tanA2 8.\ \cot A= \frac {1- \tan^2 \frac A2}{2 \tan \frac A2}

Double angle formula:

1. sin2A=2sinAcosA=2tanA1+tan2A 1.\ \sin 2A=2\sin A \cos A = \frac {2 \tan A}{1+ \tan^2 A}
2. cos2A=cos2A–sin2A=1–tan2A1+tan2A=2cos2A–1=1–2sin2A 2.\ \cos 2A=\cos^2 A-\sin^2 A= \frac {1- \tan^2 A}{1+ \tan^2 A} = 2 \cos^2 A-1=1-2\sin^2 A
3. tan2A=2tanA1–tan2A 3.\ \tan 2A= \frac {2 \tan A}{1- \tan^2 A}
4. cot2A=cot2A–12cotA 4.\ \cot 2A= \frac {\cot^2 A-1}{2 \cot A}

Multiple angle formula:

1. sin3A=3sinA–4sin3A 1.\ \sin 3A = 3 \sin A – 4 \sin^3A
2. cos3A=4cos3A–3cosA 2.\ \cos 3A = 4 \cos^3A – 3 \cos A
3. tan3A=(3tanA–tan3A)(1–3tan2A) 3.\ \tan 3A = \frac {(3 \tan A – \tan^3 A)}{(1-3\tan^2 A)}
4. cot3A=(cot3A–3cotA)(3cot2A–1) 4.\ \cot 3A= \frac {(\cot^3A – 3\cot A)}{( 3\cot^2A – 1)}
5. sin4A=4sinAcosA−8sin3AcosA 5.\ \sin 4A = 4 \sin A \cos A − 8 \sin^3 A \cos A
6. cos4A=8cos4A−8cos2A+1 6.\ \cos 4A = 8 \cos^4 A − 8 \cos^2 A + 1
7. tan4A=(4tanA–4tan3A)(1−6tan2A+tan4A) 7.\ \tan 4A =\frac {(4 \tan A – 4\tan^3 A)}{(1 − 6 \tan^2 A + \tan^4 A)}
8. cot4A=(1−6tan2A+tan4A)(4tanA–4tan3A) 8.\ \cot 4A = \frac {(1 − 6 \tan^2 A + \tan^4 A)}{(4 \tan A – 4\tan^3 A)}
9. sin5A=5sinA–20sin3A+16sin5A 9.\ \sin 5A = 5\sin A – 20 \sin^3A + 16 \sin^5 A
10. cos5A=16cos5A–20cos3A+5cosA 10.\ \cos 5A = 16 \cos^5 A – 20 \cos^3 A + 5\cos A
11. tan5A=(tan5A–10tan3A+5tanA)(1−10tan2A+5tan4A) 11.\ \tan 5A =\frac {(\tan^5 A – 10\tan^3 A + 5 \tan A)}{(1 − 10 \tan^2 A + 5\tan^4 A)}
12. cot5A=(1−10tan2A+5tan4A)(tan5A–10tan3A+5tanA) 12.\ \cot 5A = \frac {(1 − 10 \tan^2 A + 5\tan^4 A)}{(\tan^5 A – 10\tan^3 A + 5 \tan A)}

Example:

If sin A = 3/5 then find the value of sin 3A

Solution:

sin3A=335–4353 \sin 3A = 3 \left( \frac 35 \right) – 4 \left( \frac 35 \right)^3
=95–427125 = \frac 95 – 4 \left( \frac {27}{125} \right)
=95–108125 = \frac 95 – \frac {108}{125}
=95×2525–108125 = \frac 95 \times \frac {25}{25} – \frac {108}{125}
=225125–108125 = \frac {225}{125} – \frac {108}{125}
=(225–108)125 = \frac {(225-108)}{125}
=117125 = \frac {117}{125}

Example:

Find the value of Sin 30 degrees using the sine half-angle formula

Solution:

$$ \sin \frac A2 = \pm \sqrt \frac{(1- \cos A)}{2} $$
$$ \sin \frac {60}{2} = \pm \sqrt \frac{(1- \cos 60^\circ)}{2} $$
$$ \sin 30^\circ = \pm \sqrt \frac{1- \frac 12}{2} $$
$$ \sin 30^\circ = \pm \sqrt \frac 14 $$
sin30°=0.5 \sin 30^\circ = 0.5

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