Trigonometry Questions for Competitive Exams Part 13th

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.

Q25. If tan15⁰ = (2 - √3), then find the value of tan15⁰ . Cot75⁰ +  tan75⁰ . Cot15⁰ =?

Sol.  Given :-  tan15⁰ = (2 - √3)

tan(90⁰ - 75⁰) = (2 - √3)

We know that  tan(90⁰ - θ) = Cotθ, therefore we get,

Cot75⁰ = (2 - √3)

We also know that tanθ = 1/Cotθ, so we can get,

tan75⁰ = 1/ Cot75⁰

tan75⁰ = 1/(2 - √3)

On rationalizing we get,

tan75⁰ = (2 + √3)/{(2 - √3).( 2 + √3)}

On simplifying we get,

tan75⁰ = (2 + √3)

Similarly we can get,

Cot15⁰ = (2 + √3)

Now tan15⁰ . Cot75⁰ +  tan75⁰ . Cot15⁰

Putting all the values we get

= (2 - √3) . (2 - √3) + (2 + √3) . (2 + √3)

= (2 - √3)² + (2 + √3)²

We know that (a + b)² + (a - b)² = 2 {a² + b²} so we get

= 2 {2² + (√3)²}

= 2 {4 + 3} = 14

Hence the required answer is 14.

Q26. Find the value of Sin75⁰

Sol. Sin75⁰

= Sin(45⁰ + 30⁰)

We know that Sin(A + B) = SinA . CosB + CosA . SinB, on applying this formula;

= Sin45⁰ . Cos30⁰ + Cos45⁰ . Sin30⁰

= (1/√2) . (√3/2) + (1/√2) . (1/2)

= (√3/2√2) + (1/2√2)

= (√3 + 1)/2√2

 Hence the required answer Sin75⁰ = (√3 + 1)/2√2.

Q27. Find the value of Sin22.5⁰ .

Sol. Given :-  Sin22.5⁰

Let us consider 22.5⁰ = θ, then 45⁰ = 2θ

We know that Cos2θ = 1 – 2 Sin²θ, so we can write,

 Cos45⁰ = 1 – 2 Sin²22.5⁰

We know that Cos45⁰ = 1/√2, therefore

1/√2 = 1 – 2 Sin²22.5⁰

2 Sin²22.5⁰ = 1 – (1/√2)

2 Sin²22.5⁰ = (√2 - 1)/√2

Sin²22.5⁰ = (√2 - 1)/2√2

On rationalizing we get,

Sin²22.5⁰ = {√2.(√2 - 1)}/4

Taking square root both sides we get,

Sin22.5⁰ = √(2 - √2)/√4

Sin22.5⁰ = √(2 - √2)/2

Hence the required answer Sin22.5⁰ = √(2 - √2)/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........................