TRIGONOMETRIC IDENTITIES
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.
SOHCAHTOA
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
Reciprocal Trigonometric Identities
Sinθ = 1 / Cosecθ
Cscθ = 1 / Sinθ
Cosθ = 1 / Secθ
Secθ = 1 / Cosθ
Tanθ = 1 / Cotθ
Cotθ = 1 / Tanθ
Other Trigonometric Identities
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
Double Angle Identities
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
Half Angle Identities
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)
Compound Angles Identities
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]
Sum to Product Identities
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]
Triple Angle Identities
Sin3A = 3 ⋅ SinA - 4 ⋅ sin³A
Cos3A = 4 ⋅ Cos³A - 3 ⋅ Cos A
Tan3A = [3 ⋅ TanA - Tan³A] / [1 - 3 ⋅ Tan²A]
Values of Trigonometric Ratios for Standard Angles
Solving Word Problems Using Trigonometric Identities
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.
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WORD PROBLEMS
Word problems on simple equations
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Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
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