ICSE Class 10 Maths Chapter 21 Trigonometrical Identities

Unit 5: Trigonometry

Trigonometrical Identities

Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables

21.1 Trigonometry

Trigonometry means; the science which deals with the measurements of triangles.

21.2 Trigonometrical Ratios

There are six trigonometrical ratios relating to the three sides of a right-angled triangle (this has already been done by students in Class IX).

For an acute angle of a right-angled triangle:

(1) sine (sin) = \[\frac{\text{Perpendicular}}{\text{Hypotenuse}}\] => \[\sin A = \frac{BC}{AC}\]

(2) cosine (cos) = \[\frac{\text{Base}}{\text{Hypotenuse}}\] => \[\cos A = \frac{AB}{AC}\]

(3) tangent (tan) = \[\frac{\text{Perpendicular}}{\text{Base}}\] => \[\tan A = \frac{BC}{AB}\]

(4) cotangent (cot) = \[\frac{\text{Base}}{\text{Perpendicular}}\] => \[\cot A = \frac{AB}{BC}\]

(5) secant (sec) = \[\frac{\text{Hypotenuse}}{\text{Base}}\] => \[\sec A = \frac{AC}{AB}\]

(6) cosecant (cosec) = \[\frac{\text{Hypotenuse}}{\text{Perpendicular}}\] => \[\cosec A = \frac{AC}{BC}\]

Remember:

1. Each trigonometrical ratio is a real number and has no unit.

2. The values of trigonometrical ratios are always the same for the same angle.

For Example:

In right triangle ABC, \[\sin A = \frac{BC}{AC}\]

and in right triangle AMN, \[\sin A = \frac{MN}{AN}\]

Since the angle A is same for both the triangles; we have \[\sin A = \frac{BC}{AC} = \frac{MN}{AN}\]

For the same reason: \[\cos A = \frac{AB}{AC} = \frac{AM}{AN}\], \[\tan A = \frac{BC}{AB} = \frac{MN}{AM}\] and so on.

Teacher's Note

Trigonometrical ratios help us find heights of buildings and distances to objects without direct measurement, which is essential in surveying and construction work.

21.3 Relations Between Different Trigonometrical Ratios

1. Reciprocal Relations

Since \[\sin A = \frac{\text{perpendicular}}{\text{hypotenuse}}\] and \[\cosec A = \frac{\text{hypotenuse}}{\text{perpendicular}}\]

=> sin A and cosec A are reciprocals of each other

i.e. \[\sin A = \frac{1}{\cosec A}\] and \[\cosec A = \frac{1}{\sin A}\]

Similarly, (i) cos A and sec A are reciprocals of each other

i.e. \[\cos A = \frac{1}{\sec A}\] and \[\sec A = \frac{1}{\cos A}\]

(ii) tan A and cot A are reciprocals of each other

i.e. \[\tan A = \frac{1}{\cot A}\] and \[\cot A = \frac{1}{\tan A}\]

2. Quotient Relations

Since \[\sin A = \frac{\text{perpendicular}}{\text{hypotenuse}}\] and \[\cos A = \frac{\text{base}}{\text{hypotenuse}}\]

\[\therefore \frac{\sin A}{\cos A} = \frac{\text{perpendicular}}{\text{hypotenuse}} \times \frac{\text{hypotenuse}}{\text{base}}\]

\[= \frac{\text{perpendicular}}{\text{base}} = \tan A\]

Similarly, \[\frac{\cos A}{\sin A} = \cot A\]

Hence, \[\tan A = \frac{\sin A}{\cos A}\] and \[\cot A = \frac{\cos A}{\sin A}\]

3. Square Relations

In right-angled triangle ABC, with angle B = 90°;

\[\sin A = \frac{BC}{AC}\] and \[\cos A = \frac{AB}{AC}\]

\[\Rightarrow \sin^2 A + \cos^2 A = \left(\frac{BC}{AC}\right)^2 + \left(\frac{AB}{AC}\right)^2\]

\[= \frac{BC^2 + AB^2}{AC^2}\]

\[= \frac{AC^2}{AC^2} = 1\]

As, \[AB^2 + BC^2 = AC^2\]

\[\therefore \sin^2 A + \cos^2 A = 1\]

Similarly,

(i) \[1 + \tan^2 A = 1 + \left(\frac{BC}{AB}\right)^2\]

\[= \frac{AB^2 + BC^2}{AB^2} = \frac{AC^2}{AB^2}\]

\[= \left(\frac{AC}{AB}\right)^2 = \sec^2 A\]

As, \[\sec A = \frac{AC}{AB}\]

(ii) \[1 + \cot^2 A = 1 + \left(\frac{AB}{BC}\right)^2\]

\[= \frac{BC^2 + AB^2}{BC^2} = \frac{AC^2}{BC^2}\]

\[= \left(\frac{AC}{BC}\right)^2 = \cosec^2 A\]

As, \[\cosec A = \frac{AC}{BC}\]

Hence,

\[\sin^2 A + \cos^2 A = 1; \quad 1 + \tan^2 A = \sec^2 A \quad \text{and} \quad 1 + \cot^2 A = \cosec^2 A.\]

Remember:

(i) \[\sin^2 A + \cos^2 A = 1\] => \[\sin^2 A = 1 - \cos^2 A\] and \[\cos^2 A = 1 - \sin^2 A\]

(ii) \[1 + \tan^2 A = \sec^2 A\] => \[\sec^2 A - \tan^2 A = 1\] and \[\sec^2 A - 1 = \tan^2 A\]

(iii) \[1 + \cot^2 A = \cosec^2 A\] => \[\cosec^2 A - \cot^2 A = 1\] and \[\cosec^2 A - 1 = \cot^2 A\]

Teacher's Note

These fundamental relations between trigonometric ratios are the foundation for solving real-world problems involving angles and distances in engineering and physics.

21.4 Trigonometric Identities

When an equation, involving trigonometrical ratios of an angle A, is true for all values of A; the equation is called a trigonometrical identity.

Each of the relations given above; viz. reciprocal relations, quotient relations and square relations; is a trigonometrical identity.

Prove the identity: tan A + cot A = sec A . cosec A

Solution:

L.H.S. = tan A + cot A

\[= \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} = \frac{\sin^2 A + \cos^2 A}{\cos A \cdot \sin A}\]

\[= \frac{1}{\cos A \sin A}\]

As, \[\sin^2 A + \cos^2 A = 1\]

\[= \sec A \cdot \cosec A = \text{R.H.S.}\]

As, \[\sec A = \frac{1}{\cos A}\] and \[\cosec A = \frac{1}{\sin A}\]

Teacher's Note

Learning to prove identities develops logical thinking skills that are valuable in mathematics, science, and analytical problem-solving in everyday professional work.

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