Facts that Matter: Introduction to Trigonometry | Mathematics (Maths) Class 10 PDF Download

Trigonometric Ratios

In a right-angled triangle, certain ratios of the lengths of its sides depend only on the angle considered. These ratios are called trigonometric ratios. Consider triangle ABC with ∠B = 90°. For angle A, the sides are referred to as follows:

• Hypotenuse: the side opposite the right angle; it is the longest side.
• Opposite side (to ∠A): the side opposite angle A.
• Adjacent side (to ∠A): the side next to angle A (other than the hypotenuse).

The six primary trigonometric ratios for angle A are:

• sin A = (length of opposite side) ÷ (length of hypotenuse)
• cos A = (length of adjacent side) ÷ (length of hypotenuse)
• tan A = (length of opposite side) ÷ (length of adjacent side)
• cosec A = (length of hypotenuse) ÷ (length of opposite side)
• sec A = (length of hypotenuse) ÷ (length of adjacent side)
• cot A = (length of adjacent side) ÷ (length of opposite side)

The last three are simply the reciprocals of the first three:

• cosec A = 1 ÷ sin A
• sec A = 1 ÷ cos A
• cot A = 1 ÷ tan A

NOTE:

• 'sin θ' is a single symbol and the letters 'sin' cannot be detached from 'θ'. sin θ is not equal to sin × θ.
• The values of trigonometric ratios of an angle depend only on the magnitude of the angle and not on the actual size of the triangle.
• Since the hypotenuse is the longest side in a right-angled triangle, the value of sin A or cos A is always less than or equal to 1.

Trigonometric Ratios of Some Specific Angles

The commonly used specific angles are 0°, 30°, 45°, 60° and 90°. Their trigonometric ratios are given in the following table:

NOTE:

• The value of sin A increases from 0 to 1 as A increases from 0° to 90°.
• The value of cos A decreases from 1 to 0 as A increases from 0° to 90°.
• The value of tan A increases from 0 to ∞ as A increases from 0° to 90°.
• Approximate values: √2 = 1.414 and √3 = 1.732.

Trigonometric Identities

An equation involving trigonometric ratios of an angle that is true for all permissible values of the angle is called a trigonometric identity. The following are the most important identities used at this level:

• cos²A + sin²A = 1
• 1 + tan²A = sec²A
• cot²A + 1 = cosec²A

Derivation of the Pythagorean Identity

From triangle ABC with ∠B = 90°, by the Pythagorean theorem:

AB² + BC² = AC²

Divide each term by AC²:

(AB ÷ AC)² + (BC ÷ AC)² = 1

Recognise the ratios:

cos A = AB ÷ AC

sin A = BC ÷ AC

Therefore:

cos²A + sin²A = 1

Derivation of 1 + tan²A = sec²A

Start from the Pythagorean identity:

cos²A + sin²A = 1

Divide every term by cos²A:

1 + (sin²A ÷ cos²A) = 1 ÷ cos²A

Recognise the ratios:

sin ÷ cos = tan and 1 ÷ cos = sec

Thus:

1 + tan²A = sec²A

Derivation of cot²A + 1 = cosec²A

Start from the Pythagorean identity:

cos²A + sin²A = 1

Divide every term by sin²A:

(cos²A ÷ sin²A) + 1 = 1 ÷ sin²A

Recognise the ratios:

cos ÷ sin = cot and 1 ÷ sin = cosec

Therefore:

cot²A + 1 = cosec²A

Examples

Example 1 - Value of sin 30°

Using an equilateral triangle of side 2 units and dropping a perpendicular from one vertex to the opposite side gives two congruent right triangles with hypotenuse 2 and base 1.

The height of each right triangle is √(2² - 1²) = √3.

For angle 30° in the right triangle, opposite side = 1 and hypotenuse = 2.

Therefore:

sin 30° = 1 ÷ 2 = 1/2

Example 2 - Value of tan 45°

In an isosceles right triangle where the two legs are equal, opposite and adjacent sides to 45° are equal.

Therefore tan 45° = opposite ÷ adjacent = 1.

tan 45° = 1

Applications

• Trigonometric ratios are used to find heights and distances when direct measurement is difficult; measure an angle of elevation or depression and one side, then use an appropriate ratio.
• They are used in geometry, surveying, architecture, navigation and in solving many problems involving right triangles.

Summary: Trigonometric ratios relate the angles of a right-angled triangle to ratios of its sides. The six ratios - sin, cos, tan and their reciprocals cosec, sec, cot - depend only on the angle. Key identities such as cos²A + sin²A = 1 and its derived forms are essential tools for solving trigonometric problems.