Trigonometry Basics for Competitive Exams Part 7th

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.

Q6. If Cosθ + Sinθ = √2 (Cosθ), then find the value of Cosθ - Sinθ.

Sol. These types of questions are very easy to solve for the students who learnt or know the identities and formulas

Given :-  Cosθ + Sinθ = √2 (Cosθ)   ................................. eq. No. 1

Squaring both sides we get  [Cosθ + Sinθ]² = [√2 (Cosθ)]²

We know that (a + b)² = a² + b² + 2 a×b, therefore

                                                   Cos²θ + Sin²θ + 2 Sinθ × Cosθ = 2 Cos²θ

                                                   2 Sinθ × Cosθ = 2 Cos²θ – (Cos²θ + Sin²θ)

                                                   2 Sinθ × Cosθ = 2 Cos²θ – Cos²θ - Sin²θ

                                                   2 Sinθ × Cosθ = Cos²θ – Sin²θ

We also know that a² – b² = (a + b) (a - b), therefore

                                                   2 Sinθ × Cosθ = (Cosθ + Sinθ) (Cosθ - Sinθ)

From eq. No. 1, it is given that Cosθ + Sinθ = √2 (Cosθ), therefore

                                                  2 Sinθ × Cosθ = √2 (Cosθ) (Cosθ - Sinθ)

On Simplification we get,    (Cosθ - Sinθ) = √2 Sinθ.

Hence the required answer is (Cosθ - Sinθ) = √2 Sinθ.

Q7. If Sinθ + Sin²θ  = 1 then find the value of Cos²θ + Cos⁴θ.

Sol. Given :-  Sinθ + Sin²θ  = 1

                        Sinθ = 1 - Sin²θ

                        Sinθ = Cos²θ

Squaring both sides we get,

                        (Sinθ)² = (Cos²θ)²

                          Sin²θ = Cos⁴θ

We know that (Sin²θ = 1 - Cos²θ), therefore

                           1 - Cos²θ = Cos⁴θ

                           Cos²θ + Cos⁴θ = 1

Hence the required value of Cos²θ + Cos⁴θ = 1.

Q8. If Cosθ + Secθ =2 then find the value of Cos²θ + Sec²θ.

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  Secθ = 1/Cosθ hence the equation can be written as

Cosθ + (1/Cosθ) = 2 and if Cosθ = 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 :- Cosθ + Secθ =2

Squaring both sides:- (Cosθ + Secθ)² = (2)²

                                        Cos²θ + Sec²θ + 2× Cosθ × Secθ = 4

But we know that Cosθ × Secθ =1

              Therefore        Cos²θ + Sec²θ + 2× 1 = 4

                                        Cos²θ + Sec²θ = 4 – 2 =2

Hence required answer  Cos²θ + Sec²θ =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

 Cos²θ + Sec²θ =2      (because Secθ = 1/Cosθ )

Hence required answer  Cos²θ + Sec²θ =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........................