Trigonometry Basics for Competitive Exams Part 5th

Following the sequence of last post on basics of Trigonometry I am continuing the basics with their related questions in this post. Let’s prepare this to get full hold on the questions of trigonometry.

Concept 1^st :- Basic Questions based on Trigonometric Ratios:-

Q1. If 7Sin²θ + 3Cos²θ = 4 then find tanθ.

Sol. Given 7Sin²θ + 3Cos²θ = 4      ............................................ eq. No. 1

We know that        Sin²θ + Cos²θ = 1        →  Sin²θ = 1 - Cos²θ  ........... eq. No. 2

Putting the value of Sin²θ from eq (2) in eq. (1) we get

7(1 - Cos²θ) + 3Cos²θ = 4

7 – 7Cos²θ + 3Cos²θ = 4

7 – 4 = 4Cos²θ

4Cos²θ = 3

Cos²θ = ¾

Cosθ = √(3/4)

Cosθ = √3/2 = Cos30⁰

θ = 30⁰

Now tanθ = tan30⁰ = 1/√3

Hence the required value of tanθ = 1/√3

Q2. If Tanθ + Cotθ =2 then find the value of Tan²θ + Cot²θ.

Note :- These type of question are a type of question which disguise the students. Actually these questions are based on algebraic formula (x + 1/x)² = x² + 1/x² + 2

We know that  Cotθ = 1/tanθ hence the equation can be written as

Tanθ + (1/tanθ) = 2 and if tanθ = x then this equation will take the form (x + 1/x) =2

Point to Remembered :- If (x + 1/x) = 2

                                           Then xⁿ + (1/xⁿ) = 2

                                          Where n = 1, 2, 3, 4, .............. any natural number

Sol.  Method 1^st :-

Given Equation :- tanθ + Cotθ =2

Squaring both sides:- (tanθ + Cotθ)² = (2)²

                                        Tan²θ + Cot²θ + 2× tanθ × Cotθ = 4

But we know that tanθ × Cotθ =1

              Therefore        Tan²θ + Cot²θ + 2× 1 = 4

                                        Tan²θ + Cot²θ = 4 – 2 =2

Hence required answer  Tan²θ + Cot²θ =2

Method 2^nd :- As per rule:-

                                            If (x + 1/x) = 2

                                           Then xⁿ + (1/xⁿ) = 2

                                          Where n = 1, 2, 3, 4, .............. any natural number

We can answer this type of question direct verbally within second that

 Tan²θ + Cot²θ =2      (because Cotθ = 1/tanθ )

Hence required answer  Tan²θ + Cot²θ =2

 (Note you may say that this is so long but dear this is the only way to explain the whole concept of the question to you now its upto you that how fast and short handedly you can solve these type of questions)                                                                       

Hope you have enjoyed the base concept of this chapter till now. More Conceptual enjoyment will be continued for you soon, just keep visiting this blog regularly and writing me your related queries through comment or forum.

Best of Luck for Exams. To be continued soon........................