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1) The document demonstrates how to prove trigonometric identities using sum and difference formulas by relating trigonometric functions of two angles (A and B) to the functions of their sum or difference. 2) It derives the identities cos(A+B)=cosAcosB - sinAsinB and sin(A+B)=sinAcosB + cosAsinB through using distance and angle formulas in a triangle. 3) Further identities like tan(A+B), tan(A-B) are derived by substituting the expressions for sin(A+B) and cos(A+B) in the definition of tangent.
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3. Solving the problems requires knowledge of trigonometric definitions, functions, inverse functions, and trigonometric identities.
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Trigono
- 1. Proving trigonometrical identitiesusing sum and difference of two angles Vania Lundina Grade 10
- 3. Say we have this diagram: y In triangle OPQ, angle POQ = A - B P(cosA, sinA) Q(cosB, sinB) A B x 1 O P and Q are two points on the circle of radius 1 unit Coordinates of P = (cosA, sinA) Coordinates of Q = (cosB, sinB)
- 4. Now we can form an equation using: The distance formula PQ2 = (cosA-cosB)2 + (sinA-sinB)2 = cos2A – 2cosAcosB + cos2B + sin2A – 2sinAsinB + sinB2 = 2 – 2(cosAcosB + sinAsinB) The cosine formula from triangle OPQ PQ2 = 12 + 12 – 2(1)(1)cos(A-B) = 2 – 2cos(A-B)
- 5. From the two equations, we get: 2 – 2(cosAcosB + sinAsinB) = 2 – 2cos(A-B) We can cross out the “ 2 – 2 “ from the equation; 2 – 2(cosAcosB + sinAsinB) = 2 – 2cos(A-B) cos(A-B) = cosAcosB + sinAsinB
- 6. Replace B with (-B) into; cos[A-(-B) = cosAcos(-B) + sinAsin(-B) since cos(-B) = cosB and sin(-B) = -sinB, the equation can be simplified into; cos(A+B) = cosAcosB - sinAsinB
- 7. Since sin ϑ = COS (900 – ϑ) sin(A+B) = cos[900 – (A+B)] = cos[(900 – a) – B] = cos(900 – A) cosB + sin(900 – A)sinB So, sin(A+B) = sinAcosA + cosAsinB
- 8. Replace B with (-B) from the previous equation, we have; Sin(A-B) = sinAcos(-B) + cosAsin(-B) Sin(A-B) = sinAcosB-cosAsinB
- 9. tan(A+B) = sin(A+B) cos(A+B) = sinAcosB + cosAsinB cosAcosB – sinAsinB = tanA + tanB 1 – tanAtanB
- 10. Replace b with (-B), we have; tan(A-B) = tan A + tan (-B) 1 – tan A tan(-B) tan(A-B) = tanA – tanB 1 + tanAtanB
- 11. EXAmples
- 12. Simplify the equation sin 370cos 410 + cos 370 sin 410; sinAcosB + cosAsinB = sin(A+B) sin37cos 41 + cos37sin41 = sin(37 + 41) = sin 780
- 13. Simplify 1 – tan150 1 + tan150 tan A – tan B = tan (A-B) 1 + tanAtanB tan 45 – tan 15 = tan(45-15) 1 + tan 45 tan 15 1 – tan 15 = tan 300 1 + tan 15