Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Trigonometry ↓


Trigonometric Identities


Trigonometry is a branch of mathematics that studies the relationship between the sides and angles of triangles. This is a topic you will find in Class 10 Maths. One of the most fascinating parts of trigonometry is the set of rules known as trigonometric identities. These identities can simplify many algebraic expressions and solve various mathematical problems, especially those involving right-angled triangles.

Understanding trigonometric identities

At its core, a trigonometric identity is an equality that is true for all values of the angles involved — as long as both sides of the equation are defined. These identities are invaluable tools in mathematics because they allow us to rewrite trigonometric expressions in different ways.

Why are trigonometric identities important?

Trigonometric identities are useful in many mathematical situations:

  • Simplifying complex trigonometric expressions.
  • Solving trigonometric equations.
  • Proving other mathematical statements or identities.
  • Analyzing waves and oscillations in physics.
  • Engineering and computer graphics for calculations involving angles.

Basic trigonometric functions

Before diving into identities, let's recall the basic trigonometric functions based on right triangles. These include:

___________________ 
|       
|         
|         
hypotenuse 
|            
|             
|              
|               
|                
|adjacent       
opposite      | 
_________________|

The primary functions are defined as follows:

  • Sine (sin θ) = opposite/hypotenuse
  • Cosine (cos θ) = adjacent / hypotenuse
  • Tangent (tan θ) = opposite/adjacent

There are also reciprocal functions derived from these:

  • Cosecant (csc θ) = 1/sin θ = hypotenuse/opposite
  • Secant (sec θ) = 1/cos θ = Hypotenuse/Adjacent
  • Cotangent (cot θ) = 1/tan θ = adjacent/opposite

Fundamental trigonometric identities

Pythagorean identity

These are based on the Pythagorean theorem and are some of the most important identities in trigonometry:

sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = csc² θ

These equations are true for any angle θ.

Quotient identities

The quotient identities express tangent and cotangent in terms of sine and cosine:

tan θ = sin θ / cos θ
cot θ = cos θ / sin θ

Reciprocal identities

These identities relate trigonometric functions to their inverses:

sin θ = 1 / csc θ
cos θ = 1 / sec θ
tan θ = 1 / cot θ
csc θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ

Visualization of basic trigonometric identities

Let's visualize these identities with an interactive example using basic trigonometric relationships.


    
    
    
    
    sin θ
    cos θ
    1

In this unit circle diagram, sin θ and cos θ are shown. The hypotenuse is the radius of the circle - this is always 1 in our unit circle, which makes calculations simple.

Working with identities: Examples

Let's look at how we can use these identities with specific examples:

Example 1: Simplifying trigonometric expressions

Consider the expression: 1 - sin² θ using the Pythagorean identity:

sin² θ + cos² θ = 1

We can rearrange it to the following:

cos² θ = 1 - sin² θ

Thus, 1 - sin² θ = cos² θ

Example 2: Proving an identity

Prove that tan θ * sec θ = sin θ.

Start with the left side of the equation and divide it by:

tan θ * sec θ = (sin θ / cos θ) * (1 / cos θ) = sin θ / cos² θ

Now consider the Pythagorean identity:

cos² θ = 1 - sin² θ

so:

sin θ / cos² θ = sin θ

So, we conclude that tan θ * sec θ = sin θ. The identity is verified!

Example 3: Solving trigonometric equations

We now solve a simple trigonometric equation using identities. Solve sec θ - 1 = 0 for θ.

Start by expressing sec θ in terms of cos θ:

sec θ = 1 / cos θ

So, the equation becomes:

1 / cos θ - 1 = 0

which implies:

1 / cos θ = 1

This equation is as follows:

cos θ = 1

Thus, θ can be 0° or 360°, or any multiple of 360°, where the cosine function equals 1.

Practice problems

Try solving these problems using trigonometric identities:

  1. Simplify: syn² θ + cos² θ + tan² θ / sec² θ
  2. Solve for θ: tan² θ - sec² θ = -1
  3. Prove: csc² θ - cot² θ = 1
  4. Simplify: 1 + cot² θ / csc² θ
  5. If 2sin θ cos θ = sin 2θ then find θ

Conclusion

Trigonometric identities are powerful mathematical tools that allow for the simplification of expressions, the solving of equations, and the transformation of trigonometric functions. Understanding and using these identities can greatly enhance your problem-solving abilities in both mathematics and the applied sciences. As you continue to study these identities, practice regularly and challenge yourself with a variety of problems to build a solid understanding of trigonometry.


Grade 10 → 4.2


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Values of Trigonometric Ratios at Specific Angles
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Trigonometric Ratios of Complementary Angles

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