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Trigonometrical Identities Notes

Introduction

Trigonometry is like a treasure map for understanding triangles! It’s a fascinating branch of mathematics that unlocks the secrets of angles and sides, helping us measure and explore the world around us. Whether you're figuring out the height of a tree or the angle of a rocket's trajectory, trigonometry is your trusty guide. In this chapter, we dive into the exciting world of trigonometrical identities, complementary angles, and the use of four-figure tables to make calculations a breeze. Let’s embark on this mathematical adventure with enthusiasm!

Trigonometry

  • Trigonometry is the study of measuring triangles, focusing on the relationships between their sides and angles.
  • It helps us calculate unknown sides or angles in triangles, especially right-angled ones.
  • Example: In a right-angled triangle, if you know one angle and one side, trigonometry can help find the other sides or angles.

Trigonometrical Ratios

  • Trigonometrical ratios are special ratios that connect the sides of a right-angled triangle to its acute angles.
  • There are six main ratios: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec).
  • These ratios are defined for an acute angle A in a right-angled triangle with sides labeled as:
    Trigonometrical Identities Chapter Notes | Mathematics Class 10 ICSE
    • Perpendicular (BC): The side opposite to angle A.
    • Base (AB): The side adjacent to angle A.
    • Hypotenuse (AC): The longest side, opposite the right angle.
  • Formulas:
    • sin A = Perpendicular / Hypotenuse = BC / AC
    • cos A = Base / Hypotenuse = AB / AC
    • tan A = Perpendicular / Base = BC / AB
    • cot A = Base / Perpendicular = AB / BC
    • sec A = Hypotenuse / Base = AC / AB
    • cosec A = Hypotenuse / Perpendicular = AC / BC
  • Key points:
    • Each ratio is a unitless real number.
    • The value of a ratio remains constant for a given angle, regardless of the triangle’s size.
  • Example: In triangle BAC, if angle C is equal to angle F in triangle EDF, then sin C = Sin F = BA / BC = ED/ EF, showing that the sine of equal angles is consistent across different triangles.Trigonometrical Identities Chapter Notes | Mathematics Class 10 ICSE

Relations Between Different Trigonometrical Ratios

Trigonometrical ratios are interconnected through specific relationships: reciprocal, quotient, and square relations.

1. Reciprocal Relations

  • Some ratios are inverses of each other.
  • Formulas:
    • sin A = 1 / cosec A and cosec A = 1 / sin A
    • cos A = 1 / sec A and sec A = 1 / cos A
    • tan A = 1 / cot A and cot A = 1 / tan A
  • Example: If sin A = 0.5, then cosec A = 1 / 0.5 = 2.

2. Quotient Relations

  • These show how sine and cosine relate to tangent and cotangent.
  • Formulas:
    • tan A = sin A / cos A
    • cot A = cos A / sin A
  • Example: If sin A = 3/5 and cos A = 4/5, then tan A = sin A / cos A = (3/5) / (4/5) = 3/4.

3. Square Relations

  • These relations use the Pythagorean theorem to connect the ratios.Trigonometrical Identities Chapter Notes | Mathematics Class 10 ICSE
  • Formulas:
    • sin2 A + cos2 A = 1
    • 1 + tan2 A = sec2 A
    • 1 + cot2 A = cosec2 A
  • Derived forms:
    • sin2 A = 1 - cos2 A
    • cos2 A = 1 - sin2 A
    • sec2 A - tan2 A = 1
    • sec2 A - 1 = tan2 A
    • cosec2 A - cot2 A = 1
    • cosec2 A - 1 = cot2 A
  • Example: In triangle ABC, sin A = BC / AC, cos A = AB / AC. Using Pythagorean theorem (AB2 + BC2 = AC2), we get sin2 A + cos2 A = (BC2 + AB2) / AC2 = AC2 / AC2 = 1.
    Trigonometrical Identities Chapter Notes | Mathematics Class 10 ICSE

Trigonometric Identities

  • A trigonometric identity is an equation involving trigonometric ratios that holds true for all values of the angle.
  • Reciprocal, quotient, and square relations are examples of identities.
  • To prove an identity:
    • Start with the more complex side (LHS or RHS).
    • Use known trigonometric relations to simplify it to match the other side.
    • If both sides are complex, simplify both independently to a common result.

Example: Prove tan A + cot A = sec A · cosec A 

  • LHS = tan A + cot A = (sin A / cos A) + (cos A / sin A)
  • = (sin2 A + cos2 A) / (cos A · sin A) = 1 / (cos A · sin A)
  • = (1 / cos A) · (1 / sin A) = sec A · cosec A = RHS
  • Example: Prove cos4 A - sin4 A = 2 cos2A - 1 
  • LHS = (cos2 A)2 - (sin2 A)2 = (cos2 A - sin2 A)(cos2 A + sin2 A)
  • = (cos2 A - sin2 A) · 1 = cos2 A - (1 - cos2 A)
  • = cos2 A - 1 + cos2 A = 2 cos2 A - 1 = RHS

Example: Prove (1 + cot A)2+ (1 - ascend desc="Example: Prove sin A / (1 + cos A) + (1 + cos A) / sin A = 2 cosec A 

  • LHS = [sin2 A + (1 + cos A)2] / [(1 + cos A) sin A]
  • = [sin2 A + 1 + cos2 A + 2 cos A] / [(1 + cos A) sin A]
  • = [1 + 2 cos A] / [(1 + cos A) sin A] = 2 / sin A = 2 cosec A = RHS
  • Alternative method for (1 + cot A)2 + (1 - cot A)2 = 2 cosec2A:
    • LHS = 1 + cot2 A + 2 cot A + 1 + cot2 A - 2 cot A
    • = 2 + 2 cot2 A = 2 (1 + cot2 A) = 2 cosec2 A = RHS

Trigonometrical Ratios of Complementary Angles

  • Complementary angles sum to 90° (e.g., A and 90° - A).
  • The trigonometric ratios of complementary angles are related.
  • Formulas:
    • sin (90° - A) = cos A
    • cos (90° - A) = sin A
    • tan (90° - A) = cot A
    • cot (90° - A) = tan A
    • sec (90° - A) = cosec A
    • cosec (90° - A) = sec A

Example: Given cos 38° sec (90° - 2A) = 1, find A. 

  • cos 38° sec (90° - 2A) = 1
  • sec (90° - 2A) = cosec 2A, so cos 38° · (1 / sin 2A) = 1
  • sin 2A = cos 38° = cos (90° - 52°) = sin 52°
  • Thus, 2A = 52°, so A = 26°.

Using the Trigonometrical Tables

  • Trigonometrical tables provide values of sine, cosine, and tangent for angles to four decimal places.
  • Table structure:
    • Left column: degrees from 0° to 89°.
    • Middle columns: minutes (0’, 6’, 12’, ..., 54’).
    • Right columns: additional minutes (1’, 2’, 3’, 4’, 5’).
  • Note: 1° = 60 minutes (60’).
  • Example: To find a ratio (e.g., sin 36° 51’):Trigonometrical Identities Chapter Notes | Mathematics Class 10 ICSE
    • Locate the closest lower angle (36° 48’ = 0.5990).
    • Add the difference for 3’ (0.0007).
    • Result: sin 36° 51’ = 0.5990 + 0.0007 = 0.5997.
  • Example: To find an angle given a ratio (e.g., sin θ = 0.5798):Trigonometrical Identities Chapter Notes | Mathematics Class 10 ICSE
    • Find the angle with a sine just below 0.5798 (e.g., sin 35° 24’ = 0.5793).
    • Calculate difference (0.5798 - 0.5793 = 0.0005).
    • Find minutes corresponding to 0.0005 (2’).
    • Result: θ = 35° 24’ + 2’ = 35° 26’.
The document Trigonometrical Identities Chapter Notes | Mathematics Class 10 ICSE is a part of the Class 10 Course Mathematics Class 10 ICSE.
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FAQs on Trigonometrical Identities Chapter Notes - Mathematics Class 10 ICSE

1. What are the basic trigonometric ratios, and how are they defined in a right-angled triangle?
Ans. The basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). In a right-angled triangle, these ratios are defined as follows: - Sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse (sin θ = opposite/hypotenuse). - Cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse (cos θ = adjacent/hypotenuse). - Tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (tan θ = opposite/adjacent).
2. How do trigonometric identities help in solving trigonometric equations?
Ans. Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They help in solving trigonometric equations by allowing simplification and transformation of complex expressions. For example, using identities like the Pythagorean identity (sin² θ + cos² θ = 1) can simplify equations, making it easier to isolate a variable or prove an equation.
3. What are the trigonometric ratios of complementary angles?
Ans. The trigonometric ratios of complementary angles are defined as follows: If two angles are complementary (sum to 90 degrees), the sine of one angle is equal to the cosine of the other. Specifically, for angles A and B where A + B = 90 degrees: - sin A = cos B - cos A = sin B - tan A = cot B - cot A = tan B - sec A = csc B - csc A = sec B
4. How can trigonometric tables be used in calculations?
Ans. Trigonometric tables provide the values of sine, cosine, tangent, and their reciprocals for various angles. They can be used to find the trigonometric ratios without a calculator. For example, if you need to find the sine of 30 degrees, you can refer to the trigonometric table and find that sin 30° = 0.5. This is particularly useful in manual calculations and helps in verifying results obtained through other methods.
5. What is the significance of trigonometric identities in real-world applications?
Ans. Trigonometric identities are significant in various real-world applications, including physics, engineering, architecture, and computer graphics. They aid in analyzing waves, modeling periodic phenomena, designing structures, and creating realistic animations. For instance, in physics, they help in resolving forces and calculating angles in vector analysis, while in architecture, they assist in ensuring structural integrity through precise angle measurements.
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