Trigonometric Ratios

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Unit: Right Triangles

Chapter: Trigonometric Ratios

Reference: - Definition of Trigonometric Ratios, Understanding the Right Triangle in Trigonometry, Sine Ratio in Right Triangles, Cosine Ratio in Right Triangles, Tangent Ratio in Right Triangles, Reciprocal Trigonometric Ratios, Application of Trigonometric Ratios in Problem-Solving, Complementary Angles and Trigonometric Ratios, Angle-Based Relationships in Right Triangles, Use of Trigonometric Ratios in Coordinate Geometry

After studying this chapter, you should be able to understand:

• Definition of Trigonometric Ratios & Understanding the Right Triangle in Trigonometry
• Sine, Cosine & Tangent Ratio in Right Triangles
• Reciprocal & Application of Trigonometric Ratios
• Use of Trigonometric Ratios in Coordinate Geometry

Definition of Trigonometric Ratios – Trigonometric ratios represent the fundamental relationships between the sides and angles of a right triangle. These ratios allow for the analysis of angles and distances in geometric figures and provide a framework for solving problems involving right triangles.

Understanding the Right Triangle in Trigonometry – A right triangle consists of one angle that is exactly 90 degrees. The other two angles are acute, and their sum must be 90 degrees. The three sides of the triangle—hypotenuse, opposite, and adjacent—form the basis for defining trigonometric relationships.

Sine Ratio in Right Triangles – The sine ratio is a measure of how an angle in a right triangle relates to the two non-hypotenuse sides. It provides insight into the way the opposite side changes concerning the hypotenuse when the angle varies. This ratio remains consistent for a given angle regardless of the triangle's size.

Cosine Ratio in Right Triangles – The cosine ratio represents how a specific acute angle in a right triangle relates to the hypotenuse and one of the non-hypotenuse sides. It is particularly useful in problems where the length of the adjacent side is required for determining distances or structural alignments.

Tangent Ratio in Right Triangles – The tangent ratio establishes a relationship between the two shorter sides of a right triangle in relation to a given angle. It is widely used in real-world applications, including navigation and slope calculations, as it helps compare vertical and horizontal changes.

Reciprocal Trigonometric Ratios – The reciprocal trigonometric ratios—cosecant, secant, and cotangent—offer alternative representations of the sine, cosine, and tangent ratios. These relationships allow for flexibility in mathematical problem-solving and provide additional tools for analysing geometric figures.

Application of Trigonometric Ratios in Problem-Solving – Trigonometric ratios are essential for solving various practical problems, such as determining the height of objects, calculating distances between points, and understanding structural angles. These ratios extend beyond pure mathematics and are used in engineering, physics, and astronomy.

Complementary Angles and Trigonometric Ratios – Complementary angles have a special connection in trigonometry, as the ratio values of one angle correspond to those of its complement. This relationship is crucial in simplifying trigonometric expressions and understanding how different angles interact in a right triangle.

Angle-Based Relationships in Right Triangles – The values of trigonometric ratios depend solely on the magnitude of the angle, rather than the specific dimensions of the triangle. This principle enables the use of standardized trigonometric tables and functions to determine unknown measurements in various geometric applications.

Use of Trigonometric Ratios in Coordinate Geometry – Trigonometric concepts extend beyond right triangles into coordinate geometry, where they assist in analysing slopes, determining angles of elevation and depression, and connecting algebraic and geometric principles. These applications play a fundamental role in advanced geometric and analytical studies.

Trigonometry:

Introduction to Trigonometry

When we break the word “Trigonometry” we get, “tri” meaning three, “gon” meaning sides and “metry” meaning measure. So, trigonometry is a branch of mathematics that deals with the relationship between the sides and angles of a triangle.

These concepts are used by the astronomers to study the stars, used by engineers in various applications ranging from constructions to auto designs, and so on.

For example, from the picture below, we can determine the height of a building, one we know the point of viewing and the distance of view point from the base of the respective building,

We also use trigonometry in construction as well, such as determining the angle of roof, and accordingly determining the trusses or run vs rise length. Just like two above, we have applications of trigonometry far and wide such as, marine engineering, oceanography, space, video games, projectile motions (use in defense) etc.

Trigonometric Ratios and Angles:

The relationship between angles and sides is established through ratios. It is applicable only when there is a right triangle. The angle θ

 is always acute. Let us look at a right triangle and some ratios to build our understanding.

 

In ∆ QRS side QR is the side opposite to ∠ S

 and side RS is called the side adjacent to ∠S .

The following are the trigonometric ratios of ∠S

 of  ∆QRS :

1. Sine of θ

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    is written as sinθ

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   .

sinθ= Side opposite of angle θ hypotenuse= QRQS

1. Cosine of θ

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    is written as cosθ

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   .

Cos θ

 = Side adjacent of angle 

 

1. Tangent of θ

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    is written as tanθ

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   .

tan θ

 = Side opposite of angle

1. Cotangent of θ

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    is written as cotθ

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   .

cot θ

 = Side adjacent of angle θ

1. Secant θ

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    is written as sec θ

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   .

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2. Cosecant θ

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    is written as cosecθ

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   .

Cosec θ

 = hypotenuseSide opposite of angle θ= QSQR

A good and common way to remember the first three is the acronym SOH-CAH-TOA. Each triplet of the acronym represents a different equation:

SOH: Sine = Opposite ÷ Hypotenuse

CAH: Cosine = Adjacent ÷ Hypotenuse

TOA: Tangent = Opposite ÷ Adjacent

All you need to have memorized are the first three as long as you have the reciprocal relations memorized.

RECIPROCAL RELATION

1. Sin θ

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    and Cosec θ

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    are reciprocal of each other,

As, sinθ

 = QRQS   and  cosecθ  = QSQR

And on multiplying, we get

sinθ ×cosecθ= QRQS × QSQR=1

1. Cos θ

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    and Sec θ

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    are reciprocal of each other,

As, cosθ

 = RSQS   and  secθ  = QSRS

And on multiplying, we get

cosθ ×secθ= RSQS= × QSRS=1

1. Tan θ

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    and Cot θ

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    are reciprocal of each other,

As, tanθ

 = QRRS  and cotθ  = RSQR

And on multiplying, we get

tanθ ×cotθ= QRRS × RSQR=1

Remark: Note that the symbol sin θ

  is used as an abbreviation for ‘the

Sine of the angle θ

’. Sin θ  is not the product of ‘sin’ and θ . ‘sin’ separated from θ  has no meaning. Similarly, cos θ  is not the product of ‘cos’ and θ . Similar interpretations follow for other trigonometric ratios also.

Now, if we take a point P on the hypotenuse AC or a point Q on AC extended, of the right triangle ABC and draw PM perpendicular to AB and QN perpendicular to AB extended (see Fig. below), how will the trigonometric ratios of ∠

 A in ∆  PAM differ from those of ∠  A in ∆  CAB or from those of ∠ A in ∆  QAN?

To answer this, first look at these triangles. Is ∆

 PAM similar to  ∆   CAB? From properties of similar triangle, recall the AA similarity criterion. Using the criterion, you will see that the triangles PAM and CAB are similar. Therefore, by the property of similar triangles, the corresponding sides of the triangles are proportional.

So we have,

AMAB

 =APAC  =MPBC

From this, we find

MPAP

 =BCAC  = sin A.

AMAP

 =ABAC  = cos A.

MPAM

 =BCAB  = tan A. and so on.

From our observations, it is now clear that the values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.

Note: For the sake of convenience, we may write sin2A, cos2A, etc., in place of

(Sin A)², (cos A)², etc., respectively. But cosec A = (sin A)^–1 ≠

 sin^–1 A (it is called sine inverse A). sin^–1 A has a different meaning.

Similar conventions hold for the other trigonometric ratios as well. Sometimes, the Greek letter θ

 (theta) is also used to denote an angle.

QUOTIENT RELATION

Fundamental Role in Right Triangle Geometry – Trigonometric ratios provide a foundational approach to understanding the relationships between angles and side lengths in right triangles, making them essential in geometric analysis.

Interdependence of Trigonometric Ratios – The sine, cosine, and tangent ratios, along with their reciprocal counterparts, are interconnected and allow for flexible problem-solving across various geometric scenarios.

Significance in Real-World Applications – Trigonometric ratios are widely used in navigation, architecture, engineering, and physics, demonstrating their relevance beyond theoretical mathematics.

Consistency Across Different Triangles – The values of trigonometric ratios depend only on the angles of a right triangle, making them universally applicable regardless of the triangle’s size.

Extension to Advanced Mathematical Concepts – The study of trigonometric ratios serves as a stepping stone to more advanced topics, including the unit circle, trigonometric functions, and their applications in coordinate geometry and calculus.