CBSE Class 10 Introduction to Trigonometry Important Formulas and concepts for exams

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CLASS X: CHAPTER - 8

INTRODUCTION TO TRIGONOMETRY

IMPORTANT FORMULAS & CONCEPTS

The word 'trigonometry' is derived from the Greek words 'tri' (meaning three), 'gon' (meaning sides) and 'metron' (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle.

Trigonometric Ratios (T - Ratios) of an acute angle of a right triangle
In XOY-plane, let a revolving line OP starting from OX, trace out ∠XOP = θ.
From P (x, y) draw PM ⊥ to OX.
In right angled triangle OMP, OM = x (Adjacent side); PM = y (opposite side); OP = r (hypotenuse).

sin \(\theta\) = \(\frac{\text{Opposite side}}{\text{Hypotenuse}}\) = \(\frac{y}{r}\)
cos \(\theta\) = \(\frac{\text{Adjacent Side}}{\text{Hypotenuse}}\) = \(\frac{x}{r}\)
tan \(\theta\) = \(\frac{\text{Opposite side}}{\text{Adjacent Side}}\) = \(\frac{y}{x}\)
cosec \(\theta\) = \(\frac{\text{Hypotenuse}}{\text{Opposite side}}\) = \(\frac{r}{y}\)
sec \(\theta\) = \(\frac{\text{Hypotenuse}}{\text{Adjacent Side}}\) = \(\frac{r}{x}\)
cot \(\theta\) = \(\frac{\text{Adjacent Side}}{\text{Opposite side}}\) = \(\frac{x}{y}\)

Reciprocal Relations
sin \(\theta\) = \(\frac{1}{\text{cosec }\theta}\)      cosec \(\theta\) = \(\frac{1}{\text{sin }\theta}\)
cos \(\theta\) = \(\frac{1}{\text{sec }\theta}\)      sec \(\theta\) = \(\frac{1}{\text{cos }\theta}\)
tan \(\theta\) = \(\frac{1}{\text{cot }\theta}\)      cot \(\theta\) = \(\frac{1}{\text{tan }\theta}\)

Quotient Relations
tan \(\theta\) = \(\frac{\text{sin }\theta}{\text{cos }\theta}\)      and      cot \(\theta\) = \(\frac{\text{cos }\theta}{\text{sin }\theta}\)

Remark 1: sin \(\theta\) is read as the "sine of angle \(\theta\)" and it should never be interpreted as the product of 'sin' and '\(\theta\)'

Remark 2: Notation : \((\text{sin }\theta)^2\) is written as \(\text{sin}^2\theta\) (read "sin square \(\theta\)") Similarly \((\text{sin }\theta)^n\) is written as \(\text{sin}^n\theta\) (read "sin nth power \(\theta\)" ), n being a positive integer.

Note: \((\text{sin }\theta)^2\) should not be written as \(\text{sin }\theta^2\) or as \(\text{sin}^2\theta^2\)

Remark 3: Trigonometric ratios depend only on the value of \(\theta\) and are independent of the lengths of the sides of the right angled triangle.

Trigonometric ratios of Complementary angles.
sin(90° - \(\theta\)) = cos \(\theta\)
cos(90° - \(\theta\)) = sin \(\theta\)
tan(90° - \(\theta\)) = cot \(\theta\)
cot(90° - \(\theta\)) = tan \(\theta\)
sec(90° - \(\theta\)) = cosec \(\theta\)
cosec(90° - \(\theta\)) = sec \(\theta\)

Teacher's Note:
a) Complementary angles are two angles whose sum is 90°.
b) Each trigonometric ratio is equal to its co-ratio of the complementary angle.
c) These formulas are very useful in simplifying expressions where angle sums are 90°.

Trigonometric ratios for angle of measure 0°, 30°, 45°, 60° and 90° in tabular form.

Teacher's Note:
a) Memorizing standard angle values is essential for solving trigonometry problems quickly.
b) Division by zero results in values that are not defined.
c) Notice that the values of sine from 0° to 90° are the reverse of cosine values.

TRIGONOMETRIC IDENTITIES
An equation involving trigonometric ratios of an angle is said to be a trigonometric identity if it is satisfied for all values of \(\theta\) for which the given trigonometric ratios are defined.
Identity (1) : \(\sin^{2}\theta + \cos^{2}\theta = 1\)
\(\Rightarrow \sin^{2}\theta = 1 - \cos^{2}\theta\) and \(\cos^{2}\theta = 1 - \sin^{2}\theta\).
Identity (2) : \(\sec^{2}\theta = 1 + \tan^{2}\theta\)
\(\Rightarrow \sec^{2}\theta - \tan^{2}\theta = 1\) and \(\tan^{2}\theta = \sec^{2}\theta - 1\).
Identity (3) : \(\csc^{2}\theta = 1 + \cot^{2}\theta\)
\(\Rightarrow \csc^{2}\theta - \cot^{2}\theta = 1\) and \(\cot^{2}\theta = \csc^{2}\theta - 1\).

Teacher's Note:
a) Identities are true for all valid angles in their domain.
b) These three fundamental identities are derived directly from Pythagoras theorem.
c) Rearranging identities helps in substituting expressions during proofs and simplifications.

SOME TIPS

Teacher's Note:
a) SOH-CAH-TOA is a handy mnemonic for remembering right-triangle ratios.
b) The coordinate system method extends trigonometric definitions beyond 90 degrees.
c) Side definitions depend entirely on the reference angle chosen in the right triangle.

Each trigonometric function in terms of the other five. [1 Mark]

Note: csc θ is same as cosec θ.

Answer:
1. All six trigonometric ratios can be expressed in terms of any single trigonometric ratio using fundamental identities.
2. The fundamental relations used include \(\sin^2\theta + \cos^2\theta = 1\), \(1 + \tan^2\theta = \sec^2\theta\), and \(1 + \cot^2\theta = \csc^2\theta\), along with reciprocal relations.

Teacher's Note:
a) This conversion table is extremely useful for proving trigonometric identities and solving complex conditional problems.
b) The positive and negative signs account for the quadrant in which the angle θ lies.