Trigonometric Identities :
In mathematics, trigonometric identities are equalities where we would have trigonometric functions and they would be true for every value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles.
Sin θ = Opposite side / Hypotenuse side
Cos θ = Adjacent side / Hypotenuse side
Tan θ = Opposite side / Adjacent side
Csc θ = Hypotenuse side / Opposite side
Sec θ = Hypotenuse side / Adjacent side
Cot θ = Adjacent side / Opposite side
Sinθ = 1 / Cosecθ
Cscθ = 1 / Sinθ
Cosθ = 1 / Secθ
Secθ = 1 / Cosθ
Tanθ = 1 / Cotθ
Cotθ = 1 / Tanθ
Sin²θ + Cos²θ = 1
Sin²θ = 1 - Cos²θ
Cos²θ = 1 - Sin²θ
Sec²θ - Tan²θ = 1
Sec²θ = 1 + Tan²θ
Tan²θ = Sec²θ - 1
Csc²θ - Cot²θ = 1
Csc²θ = 1 + Cot²θ
Cot²θ = Csc²θ - 1
Sin2A = 2 ⋅ SinA ⋅ CosA
Cos2A = Cos²A - Sin²A
Tan2A = 2 ⋅ TanA / (1 - Tan²A)
Cos2A = 1 - 2 ⋅ Sin²A
Cos2A = 2 ⋅ Cos²A - 1
Sin2A = 2 ⋅ TanA / (1 + Tan²A)
Cos2A = (1 - Tan²A) / (1 + Tan²A)
Sin²A = (1 - Cos2A) / 2
Cos²A = (1 + Cos2A) / 2
SinA = 2 ⋅ Sin(A/2) ⋅ Cos(A/2)
CosA = Cos²(A/2) - Sin²(A/2)
TanA = 2 ⋅ Tan(A/2) / [1 - Tan²(A/2)]
CosA = 1 - 2 ⋅ Sin²(A/2)
CosA = 2 ⋅ Cos²(A/2) - 1
SinA = 2 ⋅ Tan(A/2) / [1 + Tan²(A/2)]
CosA = [1 - Tan²(A/2)] / [1 + Tan²(A/2)]
Sin²A/2 = (1 - Cos A) / 2
Cos²A/2 = (1 + Cos A) / 2
Tan²(A/2) = (1 - CosA) / (1 + CosA)
Sin(A + B) = SinA ⋅ CosB + CosA ⋅ SinB
Sin(A + B) = SinA ⋅ CosB + CosA ⋅ SinB
Cos(A + B) = CosA ⋅ CosB - SinA ⋅ SinB
Cos(A - B) = CosA ⋅ CosB + SinA ⋅ SinB
Tan(A + B) = [TanA + TanB] / [1- TanA ⋅ TanB]
Tan(A - B) = [TanA - TanB] / [1 + TanA ⋅ TanB]
SinC + SinD = 2 ⋅ Sin[(C+D) / 2] ⋅ cos [(C-D) / 2]
SinC - SinD = 2 ⋅ Cos [(C+D) / 2] ⋅ Sin [(C-D) / 2]
CosC + CosD = 2 ⋅ Cos [(C+D) / 2] ⋅ Cos [(C-D) / 2]
CosC - CosD = 2 ⋅ Sin [(C+D) / 2] ⋅ Sin [(C-D) / 2]
Sin3A = 3 ⋅ SinA - 4 ⋅ sin³A
Cos3A = 4 ⋅ Cos³A - 3 ⋅ Cos A
Tan3A = [3 ⋅ TanA - Tan³A] / [1 - 3 ⋅ Tan²A]
Step 1 :
Understanding the question and drawing the appropriate diagram are the two most important things to be done in solving word problems in trigonometry.
Step 2 :
If it is possible, we have to split the given information. Because, when we split the given information in to parts, we can understand them easily.
Step 3 :
We have to draw diagram almost for all of the word problems in trigonometry. The diagram we draw for the given information must be correct. Drawing diagram for the given information will give us a clear understanding about the question.
Step 4 :
Once we understand the given information clearly and correct diagram is drawn, solving word problems in trigonometry would not be a challenging work.
Step 5 :
After having drawn the appropriate diagram based on the given information, we have to give name for each position of the diagram using English alphabets (it is clearly shown in the word problem given below). Giving name for the positions would be easier for us to identify the parts of the diagram.
Step 6 :
Now we have to use one of the three trigonometric ratios (sin, cos and tan) to find the unknown side or angle.
Once the diagram is drawn and we have translated the English Statement (information) given in the question as mathematical equation using trigonometric ratios correctly, 90% of the work will be over. The remaining 10% is just getting the answer. That is solving for the unknown.
These are the most commonly steps involved in solving word problems in trigonometry.
After having gone through the stuff given above, we hope that the students would have understood trigonometric identities.
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