Trigonometry Questions for Competitive Exams Part 11th

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.

Q19. If tanθ =3/4 and θ is an acute angle, find the value of cosecθ.

Sol.  Given :-  tanθ =3/4

In triangle ABC with 90⁰ at B, tanθ = AB/BC = 3/4

Let AB = 3k and BC = 4k.

On applying Pythagoras Theorem in right angled triangle:

(AC)² = (AB)² + (BC)²

(AC)² = (3k)² + (4k)²

(AC)² = 9k² + 16k²

(AC)² = 25k²

AC = 5k.

Now Cosecθ = AC/AB

Cosecθ = 5k/3k

Cosecθ = 5/3.

Hence the required answer is Cosecθ = 5/3.

Note : For time saving I would like to give you 5 important triplets, which form a right angled triangle, prepare them it will help you a lot;

3,4 and 5 (3 and 4 will be measures of base or perpendicular and 5 will be hypotenuse.)

6,8 and 10 (6 and 8 will be measures of base or perpendicular and 10 will be hypotenuse.)

12,5 and 13. (12 and 5 will be measures of base or perpendicular and 13 will be hypotenuse.)

15,8 and 17. (15 and 8 will be measures of base or perpendicular and 17 will be hypotenuse.)

24, 7 and 25. (24 and 7 will be measures of base or perpendicular and 25 will be hypotenuse.)

Q20. If (tanθ + Cotθ) = 5, then (tan²θ + Cot²θ) = ?  

Sol. Given :-  (tanθ + Cotθ) = 5

Squaring both sides we get;

(tanθ + Cotθ)² = (5)²

Applying the formula (a + b)² = a² + b² + 2ab

(tanθ)² + (Cotθ)² + 2.tanθ . Cotθ = 25

We know that tanθ . Cotθ = 1, therefore

tan²θ + Cot²θ + 2(1) = 25.

tan²θ + Cot²θ = 25 ─ 2.

tan²θ + Cot²θ = 23.

Hence the required answer tan²θ + Cot²θ = 23.

Q21. If (Cosθ + Secθ) = 5/2, then (Cos²θ + Sec²θ) = ?

Sol. Given :-  (Cosθ + Secθ) = 5/2

Squaring both sides we get;

(Cosθ + Secθ)² = (5/2)²

Applying the formula (a + b)² = a² + b² + 2ab

(Cosθ)² + (Secθ)² + 2.Cosθ . Secθ = 25/4

We know that Cosθ . Secθ = 25/4, therefore

Cos²θ + Sec²θ + 2(1) = 25/4.

Cos²θ + Sec²θ = (25/4) ─ 2.

Cos²θ + Sec²θ = 17/4.

Hence the required answer Cos²θ + Sec²θ = 17/4.

 (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........................