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Trigono

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Trigono

1. 1. Proving trigonometrical identitiesusing sum and difference of two angles<br />Vania Lundina<br />Grade 10<br />
2. 2. Basic knowledge<br />
3. 3. Say we have this diagram:<br />y<br />In triangle OPQ, <br />angle POQ = A - B<br />P(cosA, sinA)<br />Q(cosB, sinB)<br />A<br />B<br />x<br />1<br />O<br />P and Q are two points on the circle of radius 1 unit<br />Coordinates of P = (cosA, sinA)<br />Coordinates of Q = (cosB, sinB)<br />
4. 4. Now we can form an equation using:<br />The distance formula<br /> PQ2 = (cosA-cosB)2 + (sinA-sinB)2<br /> = cos2A – 2cosAcosB + cos2B + sin2A – 2sinAsinB + sinB2<br />= 2 – 2(cosAcosB + sinAsinB)<br />The cosine formula from triangle OPQ<br />PQ2 = 12 + 12 – 2(1)(1)cos(A-B)<br /> = 2 – 2cos(A-B)<br />
5. 5. From the two equations, we get:<br />2 – 2(cosAcosB + sinAsinB) = 2 – 2cos(A-B)<br />We can cross out the “ 2 – 2 “ from the equation;<br />2 – 2(cosAcosB + sinAsinB) = 2 – 2cos(A-B)<br />cos(A-B) = cosAcosB + sinAsinB<br />
6. 6. Replace B with (-B) into;<br />cos[A-(-B) = cosAcos(-B) + sinAsin(-B)<br />since cos(-B) = cosB and sin(-B) = -sinB, <br />the equation can be simplified into;<br />cos(A+B) = cosAcosB - sinAsinB<br />
7. 7. Since sin ϑ = COS (900 – ϑ)<br />sin(A+B) = cos[900 – (A+B)]<br /> = cos[(900 – a) – B]<br /> = cos(900 – A) cosB + sin(900 – A)sinB<br />So, sin(A+B) = sinAcosA + cosAsinB<br />
8. 8. Replace B with (-B) from the previous equation, we have;<br />Sin(A-B) = sinAcos(-B) + cosAsin(-B)<br />Sin(A-B) = sinAcosB-cosAsinB<br />
9. 9. tan(A+B) = sin(A+B)<br />cos(A+B)<br /> = sinAcosB + cosAsinB<br />cosAcosB – sinAsinB<br /> = tanA + tanB<br /> 1 – tanAtanB<br />
10. 10. Replace b with (-B), we have;<br />tan(A-B) = tan A + tan (-B)<br /> 1 – tan A tan(-B)<br />tan(A-B) = tanA – tanB<br /> 1 + tanAtanB<br />
11. 11. EXAmples<br />
12. 12. Simplify the equation sin 370cos 410 + cos 370 sin 410;<br />sinAcosB + cosAsinB = sin(A+B)<br />sin37cos 41 + cos37sin41 = sin(37 + 41)<br /> = sin 780<br />
13. 13. Simplify 1 – tan150<br /> 1 + tan150<br />tan A – tan B = tan (A-B)<br />1 + tanAtanB<br />tan 45 – tan 15 = tan(45-15) <br />1 + tan 45 tan 15<br />1 – tan 15 = tan 300<br />1 + tan 15<br />