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Angle transformation -Trigonometry formulas

Angle transformation

1. sin(A+B)=sinAcosB+cosAsinB 1. \ \sin(A + B) = \sin A \cos B + \cos A \sin B
2. sin(A−B)=sinAcosB−cosAsinB 2. \ \sin(A − B) = \sin A \cos B − \cos A \sin B
3. cos(A+B)=cosAcosB−sinAsinB 3. \ \cos(A + B) = \cos A \cos B − \sin A \sin B
4. cos(A−B)=cosAcosB+sinAsinB 4. \ \cos(A − B) = \cos A \cos B + \sin A \sin B
5. tan(A+B)=(tanA+tanB)(1−tanAtanB) 5. \ \tan(A + B) = \frac {(\tan A + \tan B)}{(1 − \tan A \tan B)}
6. tan(A−B)=(tanA−tanB)(1+tanAtanB) 6. \ \tan(A − B) = \frac {(\tan A − \tan B)}{(1 + \tan A \tan B)}
6. cot(A+B)=(cotA.cotB–1)(cotB+cotA) 6. \ \cot (A + B) = \frac {(\cot A.\cot B – 1)}{(\cot B + \cot A)}
6. cot(A–B)=(cotA.cotB+1)(cotB–cotA) 6. \ \cot (A – B) = \frac {(\cot A.\cot B + 1)}{(\cot B – \cot A)}

Example:

Find value ofsin15° \text{Find value of} \sin 15^\circ

Solution:

sin15°=sin(45°–30°) \sin 15^\circ= \sin (45^\circ – 30^\circ)
=sin45°cos30°–cos45°sin30° = \sin 45^\circ \cos 30^\circ – \cos 45^\circ \sin 30^\circ
=12×32–12×12 = \frac {1}{\sqrt {2}} \times \frac {\sqrt {3}}{2} – \frac {1}{\sqrt {2}} \times \frac {1}{2}
=322–122 = \frac {\sqrt {3}}{2 \sqrt {2}} – \frac {1}{2 \sqrt {2}}
=3–122 = \frac {\sqrt {3}-1}{2 \sqrt {2}}

Example:

Find value ofsin75° \text{Find value of} \sin 75^\circ

Solution:

sin15°=sin(45°+30°) \sin 15^\circ= \sin (45^\circ + 30^\circ)
=sin45°cos30°+cos45°sin30° = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ
=12×32+12×12 = \frac {1}{\sqrt {2}} \times \frac {\sqrt {3}}{2} + \frac {1}{\sqrt {2}} \times \frac {1}{2}
=322+122 = \frac {\sqrt {3}}{2 \sqrt {2}} + \frac {1}{2 \sqrt {2}}
=3+122 = \frac {\sqrt {3}+1}{2 \sqrt {2}}

 

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