# 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,

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

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.

## Recent Articles

1. ### What is a Triangle? | Types of Triangle | Scalene Triangle | Isosceles

Jun 17, 24 11:22 PM

A simple closed curve or a polygon formed by three line-segments (sides) is called a triangle. The above shown shapes are triangles. The symbol of a triangle is ∆. A triangle is a polygon with three s…

2. ### Interior and Exterior of an Angle | Interior Angle | Exterior Angle

Jun 16, 24 05:20 PM

Interior and exterior of an angle is explained here. The shaded portion between the arms BA and BC of the angle ABC can be extended indefinitely.

3. ### Angles | Magnitude of an Angle | Measure of an angle | Working Rules

Jun 16, 24 04:12 PM

Angles are very important in our daily life so it’s very necessary to understand about angle. Two rays meeting at a common endpoint form an angle. In the adjoining figure, two rays AB and BC are calle

4. ### What is a Polygon? | Simple Closed Curve | Triangle | Quadrilateral

Jun 16, 24 02:34 PM

What is a polygon? A simple closed curve made of three or more line-segments is called a polygon. A polygon has at least three line-segments.