3) Trigonometrical ratios:

In a right-angled triangle, the Pythagoras theorem states

(perpendicular )² + ( base )² = ( hypotenuse )²

There are some properties pertaining to the right-angled triangle. It is to be noted that in the following formulas, P stands for perpendicular, B stands for base and H stands for the hypotenuse.

1. SinA = P / H
2. CosA = B / H
3. TanA = P / B
4. CotA = B / P
5. CosecA = H / P
6. SecA = H/B
7. Sin²A + Cos²A = 1
8. Tan²A + 1 = Sec²A

Cot²A + 1 = Cosec²A

In order to find a relationship between various trigonometric identities, there are some important formulas:
1. TanA = SinA / CosA
2. CotA = CosA / SinA
3. CosecA = 1 / SinA
4. SecA = 1 / CosA

There are some formulas that are very crucial to solving higher level sums.

1. Sin (A +B) =SinA . CosB + CosA . SinB
2. Sin (A – B) = SinA . CosB – CosA . SinB
3. Cos (A + B) = CosA . CosB – SinA . SinB
4. Cos (A – B) = CosA. CosB + SinA . SinB
5. Tan (A + B) = TanA + TanB / 1 – TanA . TanB
6. Tan (A – B) = TanA –TanB / 1 + TanA . TanB
7. Sin ( A + B) . Sin (A – B) = Sin²A – Sin²B = Cos²B – Cos²A
8. Cos (A + B) . Cos (A – B) = Cos²A – Sin²B = Cos²B – Sin²A
9. Sin2A = 2 . SinA . CosA = 2 . TanA / (1 + Tan²A)
10. Cos2A = Cos²A – Sin²A = 1 – 2Sin²A = 2Cos²A – 1 = (1 – Tan²A) / (1 + Tan²A)
11. Tan2A = 2TanA / (1 – Tan²A)
12.  Sin3A = 3 . SinA – 4 . Sin³A
13. Cos3A = 4 . Cos³A – 3 . CosA
14. Tan3A = (3TanA – Tan³A) / (1 – 3Tan²A)
15. SinA + SinB = 2 Sin (A + B)/2 Cos (A – B)/2
16. SinA – SinB = 2 Sin (A – B)/2 Cos (A + B)/2
17. CosA + CosB = 2 Cos(A – B)/2 Cos (A + B)/2
18. CosA – CosB = 2 Sin(B – A)/2 Sin (A + B)/2
19. TanA + TanB = Sin (A + B) / CosA . CosB
20. SinA CosB = Sin (A + B) + Sin (A – B)
21. CosA SinB = Sin (A + B) – Sin (A – B)
22. CosA CosB = Cos (A + B) + Cos (A – B)
23.  SinA SinB = Cos (A – B) – Cos (A + B)

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