Trigonometric Identities

Introduction to Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for any angle. These identities are crucial in simplifying trigonometric expressions, solving equations, and proving other identities in trigonometry. Whether you’re a high school student or preparing for competitive exams, mastering trigonometric identities will help you solve complex problems quickly and efficiently.

 

Table of Contents: Trigonometric Identities

• Introduction
• List of Trigonometric Functions
  • Reciprocal Identities
  • Pythagorean Identities
  • Ratio Identities
  • Opposite, Complementary & Supplementary Identities
  • Sum & Difference Identities
  • Double & Half Angle Identities
  • Product-to-Sum Identities
  • Sum-to-Product Identities
• Trigonometric Identities of Products
• Solved Examples
• Fun Facts About Trigonometric Identities
• Common Misconceptions
• Conclusion

 

List of Trigonometric Functions

Before diving into the identities, let’s recap the list of trig functions:

 

 

These functions form the basis of trigonometric identities.

 

Reciprocal Identities

Reciprocal identities connect basic trigonometric functions with their reciprocal counterparts:

• csc⁡θ=1/sin⁡θ

• sec⁡θ=1/cos⁡θ

• cot⁡θ=1/tan⁡θ

These reciprocal identities help in converting functions and simplifying expressions.

 

Pythagorean Identities

These are derived from the Pythagorean Theorem:

• sin²(θ) + cos²(θ) = 1

• 1 + tan²(θ) = sec²(θ)

• 1 + cot²(θ) = csc²(θ)

The Pythagorean identities are widely used in trigonometric proofs and calculus.

 

Ratio Identities

Ratio identities are straightforward and relate one trigonometric function to another:

• tan(θ) = sin(θ) / cos(θ)

• cot(θ) = cos(θ) / sin(θ)

These ratio identities are handy when converting between tangent, sine, and cosine.

 

Opposite Angle Identities

Used when the angle is negative:

• sin(-θ) = -sin(θ)

• cos(-θ) = cos(θ)

• tan(-θ) = -tan(θ)

These opposite-angle identities help in handling trigonometric functions of negative angles.

 

Complementary Angles Identities

Involves angles that add up to 90° (or π/2 radians):

• sin(90° - θ) = cos(θ)

• cos(90° - θ) = sin(θ)

• tan(90° - θ) = cot(θ)

These complementary angles identities help in evaluating functions using reference angles.

 

Supplementary Angles Identities

For angles adding up to 180°:

• sin(180° - θ) = sin(θ)

• cos(180° - θ) = -cos(θ)

• tan(180° - θ) = -tan(θ)

Supplementary angles identities are useful when angles lie in the second quadrant.

 

Sum and Difference of Angles Identities

Used for evaluating trigonometric functions of the sum or difference of two angles:

• sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)

• cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)

• tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))

These sum and difference of angles identities are powerful tools in advanced trigonometry.

 

Double Angle Identities

Used when angles are doubled:

• sin(2θ) = 2sin(θ)cos(θ)

• cos(2θ) = cos²(θ) - sin²(θ)

• tan(2θ) = 2tan(θ) / (1 - tan²(θ))

Double-angle identities are frequently used in integration and differentiation.

 

Half Angle Identities

Derived from the double-angle formulas:

• sin²(θ/2) = (1 - cos(θ)) / 2

• cos²(θ/2) = (1 + cos(θ)) / 2

These half-angle identities are helpful for calculus and trigonometric simplifications.

 

Product-Sum Identities

These identities convert products of functions into sums:

• sin(A)sin(B) = ½[cos(A - B) - cos(A + B)]

• cos(A)cos(B) = ½[cos(A - B) + cos(A + B)]

• sin(A)cos(B) = ½[sin(A + B) + sin(A - B)]

Product-sum identities simplify complex trigonometric expressions in signal processing.

 

Trigonometric Identities of Products

What Are Trigonometric Identities of Products?

Trigonometric Identities of Products, also known as Product-to-Sum identities, are formulas used to convert products of trigonometric functions (like sine and cosine) into a sum or difference of angles. These identities are very helpful in simplifying trigonometric expressions and solving integrals involving trigonometric functions.

They are most commonly used in:

• Trigonometric simplifications

• Integration in calculus

• Signal processing and wave analysis

• Solving trigonometric equations
  
  

Product-to-Sum Identities

These identities convert the product of sine and cosine functions into a sum or difference form:

 

These are useful when simplifying products of trigonometric terms during problem solving or integration.

 

Sum-to-Product Formulas (Reverse Identities)

These are the reverse of the product identities and convert a sum or difference of trigonometric expressions into a product form:

 

These formulas are especially useful in transforming expressions into simpler or integrable forms.

 

Solved Examples

Example 1: Simplify cos(5x) · cos(3x)

Using the identity: cos A · cos B 

= (1/2)[cos(A − B) + cos(A + B)]

= (1/2)[cos(2x) + cos(8x)]

Example 2: Simplify sin(2x) · cos(3x)

Using the identity: sin A · cos B = (1/2)[sin(A + B) + sin(A − B)]

= (1/2)[sin(5x) + sin(−x)]
= (1/2)[sin(5x) − sin(x)]

Example 3: Simplify sin(4x) · sin(2x)

Using identity: sin A · sin B = (1/2)[cos(A − B) − cos(A + B)]

= (1/2)[cos(2x) − cos(6x)]

 

Fun Facts About Trigonometric Identities

• The word “trigonometry” comes from the Greek words trigonon (triangle) and metron (measure).

• Ancient Indian mathematician Aryabhata used early trigonometric identities in astronomy around 500 CE.

• The Pythagorean identity sin⁡2θ+cos⁡2θis one of the most commonly used identities in math and physics.
  
  

Common Misconceptions

• Misconception: sin⁡2θ=sin⁡(θ2)
  Correct: sin⁡2θ=(sin⁡θ)2
  
  
• Misconception: cos⁡(90∘−θ)=cos⁡θ
  Correct: cos⁡(90∘−θ)=sin⁡θ
  
  
• Misconception: Identities are formulas that must be memorized without understanding.
  Correct: Derive and understand identities using triangles and the unit circle; it helps you remember better.
  
  

Conclusion

Trigonometric identities play a key role in solving mathematical problems involving angles. From basic reciprocal identities to advanced product-sum identities, every identity has its unique application. Understanding and applying these identities builds a strong foundation in trigonometry and prepares students for higher-level mathematics and real-life applications in fields like architecture, physics, and engineering.

 

Related Links

Trigonometry Formula - Discover all the important trigonometric formulas in one place for quick learning and easy exam revision.

Maths Calculator - Quickly solve equations and perform calculations with our free calculator.

 

FAQs on Trigonometric Identities

1. What are the 3 main trigonometric identities?

The three main trigonometric identities, also known as Pythagorean identities, are:

• sin²θ + cos²θ = 1

• 1 + tan²θ = sec²θ

• 1 + cot²θ = cosec²θ

These are the foundational identities from which many other trigonometric formulas are derived.

 

2. What are the trigonometric identities for Class 10th?

In Class 10 Maths, the commonly used trigonometric identities include:

• sin²θ + cos²θ = 1

• 1 + tan²θ = sec²θ

• 1 + cot²θ = cosec²θ

Also, the basic trigonometric ratios for a right-angled triangle are important:

• sinθ = Opposite / Hypotenuse

• cosθ = Adjacent / Hypotenuse

• tanθ = Opposite / Adjacent

• cotθ = Adjacent / Opposite

• secθ = Hypotenuse / Adjacent

• cosecθ = Hypotenuse / Opposite

 

These are essential in solving problems related to heights, distances, and right-angled triangles.

3. What are the 8 fundamental trigonometric identities?

The 8 fundamental trigonometric identities are:

• sin²θ + cos²θ = 1

• 1 + tan²θ = sec²θ

• 1 + cot²θ = cosec²θ

• sinθ = 1 / cosecθ

• cosθ = 1 / secθ

• tanθ = sinθ / cosθ

• cotθ = cosθ / sinθ

• sec²θ − tan²θ = 1 (can be derived from identity #2)

 

4. What are the 11 identities in trigonometry?

These are often referred to as the 11 standard trigonometric identities, mostly covering reciprocal and Pythagorean identities:

1. sin²θ + cos²θ = 1

2. 1 + tan²θ = sec²θ

3. 1 + cot²θ = cosec²θ

4. sinθ = 1 / cosecθ

5. cosθ = 1 / secθ

6. tanθ = sinθ / cosθ

7. cotθ = cosθ / sinθ

8. cosecθ = 1 / sinθ

9. secθ = 1 / cosθ

10. cotθ = 1 / tanθ

11. tanθ = 1 / cotθ

 

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