Trig Ratio Problems

Basic Trig ratio problems are very important when dealing with triangles. In the below questions we will learn how to find the values of the other ratio where one ratio is given.


1. If sec θ = 17/8 and θ is a positive acute angle, find the value of csc θ using Pythagoras theorem.

Solution:

Draw a right-angled ∆ ABC such that ∠ABC = θ,

Hypotenuse = BA = 17, and Adjacent side (or base) = BC = 8.

Then we get,

sec θ = 17/8



Now, from the right-angled ∆ ABC we get,

Trig Ratio Problems

AC2 + BC2 = BA2

⇒ AC2 = BA2 - BC2

⇒ AC2 = (17)2 - 82

⇒ AC2 = 289 - 64

⇒ AC2 = 225



Therefore, AC = 15 (Since θ is a positive acute angle so, AC is also positive)

Therefore, csc θ = BA/AC

⇒ csc θ = 17/15


In this question on Trig ratio problems we will learn how to find the value of sin θ when θ is a positive acute angle.


2. If tan θ + sec θ = 2/√3 and θ is a positive acute angle, find the value of sin θ.

Solution:

Given, tan θ + sec θ = 2/√3,

⇒ sin θ/cos θ + 1/cos θ  = 2/√3,( Since tan θ = sin θ/cos θ and sec θ = 1/cos θ)

⇒ (sin θ + 1)/cos θ = 2/√3            

⇒ √3 (sin θ + 1) = 2 cos θ

⇒ 3(sin θ + 1)2 = 4 cos2 θ, (Squaring both sides)

⇒ 3(sin2 θ + 2 sin θ + 1) = 4(1 - sin2 θ)

⇒ 3 sin2 θ + 6 sin θ + 3 = 4 - 4 sin2 θ

⇒ 3 sin2 θ + 6 sin θ + 3 - 4 + 4 sin2 θ = 0

⇒ 7 sin2 θ + 6 sin θ - 1 = 0

⇒ 7 sin2 θ + 7 sin θ - sin θ - 1 =0

⇒   7 sin θ (sin θ + 1) - 1 (sin θ + 1) =0         

⇒ (7 sin θ - 1)(sin θ + 1) = 0

Therefore,

Either, 7 sin θ - 1 = 0

⇒ 7 sin θ = 1

 ⇒ sin θ = 1/7


or, sin θ + 1 = 0

⇒ sin θ = - 1

According to the problem, θ is a positive acute angle; so, we neglect, sin θ = -1.

Therefore, sin θ = 1/7 

The below solved Trig ratio problems will help us to find the values of the ratio using trigonometric identity.


3. If θ is a positive acute angle and sec θ = 25/7, find the value of csc θ using trigonometric identity.

Solution:

Given, sec θ = 25/7

Therefore, cos θ = 1/sec θ

⇒ cos θ = 1/(25/7)

⇒ cos θ = 7/25

We know that, sin2 θ + cos2 θ = 1

⇒ sin2 θ = 1 - cos2 θ

⇒ sin2 θ = 1 - (7/25)2

⇒ sin2 θ = 1 - (49/625)

⇒ sin2 θ = (625 – 49)/625

⇒ sin2 θ = 576/625

Now, taking square root on both the sides we get,

⇒ sin θ  = 24/25 (Since θ is a positive acute angle so, sin θ is also positive)

Therefore, csc θ = 1/sin θ

⇒ csc θ = 1/(24/25)

⇒ csc θ = 25/24 . 

In this question on Trig ratio problems we will learn how to find the minimum value of the given T-ratio.

4. Find the minimum value of cos2 θ + sec2 θ

Solution:

cos2 θ + sec2 θ

= (cos θ)2 + (sec θ)2 - 2 cos θ ∙ sec θ + 2 cos θ sec θ

= (cos θ - sec θ)2 + 2 ∙ 1 (since, cos θ ∙ sec θ = 1)

= (cos θ - sec θ)2 + 2

(cos θ - sec θ)2 ≥ 0

Therefore, (cos θ - sec θ)2 + 2 ≥ 2 (since, adding 2 on both the sides)

i.e., cos2 θ + sec2 θ ≥ 2

Therefore, the minimum value of cos2 θ + sec2 θ is 2.

 Trigonometric Functions





10th Grade Math

From Trig Ratio Problems to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Area, Perimeter and Volume | Square, Rectangle, Cube,Cubo

    Jul 28, 25 03:00 AM

    Volume of a Cuboids
    In this worksheet on area perimeter and volume you will get different types of questions on find the perimeter of a rectangle, find the perimeter of a square, find the area of a rectangle, find the ar…

    Read More

  2. Worksheet on Volume of a Cube and Cuboid |The Volume of a RectangleBox

    Jul 25, 25 03:15 AM

    Volume of a Cube and Cuboid
    We will practice the questions given in the worksheet on volume of a cube and cuboid. We know the volume of an object is the amount of space occupied by the object.1. Fill in the blanks:

    Read More

  3. Volume of a Cuboid | Volume of Cuboid Formula | How to Find the Volume

    Jul 24, 25 03:46 PM

    Volume of Cuboid
    Cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid will have a length, breadth and height. Hence we can conclude that volume is 3 dimensional. To measur…

    Read More

  4. Volume of a Cube | How to Calculate the Volume of a Cube? | Examples

    Jul 23, 25 11:37 AM

    Volume of a Cube
    A cube is a solid box whose every surface is a square of same area. Take an empty box with open top in the shape of a cube whose each edge is 2 cm. Now fit cubes of edges 1 cm in it. From the figure i…

    Read More

  5. 5th Grade Volume | Units of Volume | Measurement of Volume|Cubic Units

    Jul 20, 25 10:22 AM

    Cubes in Cuboid
    Volume is the amount of space enclosed by an object or shape, how much 3-dimensional space (length, height, and width) it occupies. A flat shape like triangle, square and rectangle occupies surface on…

    Read More