Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Trigonometry ↓


Trigonometric Ratios and Identities


Trigonometry is a branch of mathematics that studies relationships involving the lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies. Trigonometry is the study of triangles; with an emphasis on right triangles, where one angle is 90 degrees.

In the context of a right triangle, the primary focus is on the relationships and properties of the three main components: the sides, the angles, and the trigonometric ratios that relate them.

Understanding the basics of right-angle triangles

A right triangle is a triangle in which one angle is exactly 90 degrees, called a right angle. The side opposite the right angle is called the hypotenuse, which is also the longest side of a right triangle. The other two sides are called legs, which we often refer to as the adjacent side and opposite side relative to a given angle other than a right angle.

Visual example of a right-angled triangle

A B C Ear

In the above triangle, the angle at point B is a right angle. The line segment AC is the hypotenuse, while AB and BC are the legs of the triangle.

Trigonometric ratios

Trigonometric ratios are used to relate the angles and sides of a right triangle. These ratios are fundamental in trigonometry and include sine, cosine, and tangent. Each of these ratios is defined based on a specific angle in a right triangle.

1. Sine

The sine of an angle θ in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

sin(θ) = opposite side/hypotenuse

Example 1:

If angle θ is at vertex A, BC is the side opposite to angle θ, and AC is the hypotenuse. If BC = 5 and AC = 13, then:

sin(θ) = 5 / 13

2. Cosine

The cosine of the angle θ in a right-angled triangle is the ratio of the length of the adjacent side to the hypotenuse.

cos(θ) = adjacent side / hypotenuse

Example 2:

If the angle θ is at the vertex A, then AB is the adjacent side of the angle θ, and AC is the hypotenuse. If AB = 12 and AC = 13, then:

cos(θ) = 12 / 13

3. Tangent

The tangent of angle θ in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

tan(θ) = opposite side / adjacent side

Example 3:

If angle θ is at vertex A, BC is the opposite side and AB is the adjacent side of angle θ. If BC = 5 and AB = 12, then:

tan(θ) = 5 / 12

Other trigonometric ratios

4. Cosecant

The cosecant of angle θ is the inverse of the sine of θ.

csc(θ) = 1/sin(θ) = hypotenuse/opposite side

5. Secant

The secant of angle θ is the inverse of the cosine of angle θ.

sec(θ) = 1/cos(θ) = hypotenuse/adjacent side

6. Cotangent

The cotangent of angle θ is the inverse of the tangent of θ.

cot(θ) = 1/tan(θ) = adjacent side / opposite side

Fundamental trigonometric identities

Trigonometric identities are equations that are true for all values of the variables involved. Knowing these identities helps in solving complex trigonometric equations and understanding the relationships between different trigonometric functions.

Pythagorean identity

One of the most important trigonometric identities is the Pythagorean identity, which connects sine and cosine. This identity states:

sin²(θ) + cos²(θ) = 1

This identity follows from the Pythagorean theorem when considering a unit circle.

Reciprocal identities

Reciprocal identities are derived from the definitions of trigonometric ratios. They help express functions in terms of their reciprocals:

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

Quotient identities

The quotient identities relate tangent and cotangent to sine and cosine:

tan(θ) = sin(θ)/cos(θ) cot(θ) = cos(θ)/sin(θ)

Trigonometric ratios of specific angles

In trigonometry, specific angles are often used for which the trigonometric ratios have simple values. These include angles such as 0°, 30°, 45°, 60° and 90°.

Trigonometric ratios for 45°

If we consider a right triangle, where each non-right angle is 45°. Using the concept of an isosceles right triangle, we can determine:

sin(45°) = cos(45°) = √2/2 tan(45°) = 1

Trigonometric ratios for 30° and 60°

Consider an equilateral triangle divided into two right triangles. The ratios are as follows:

sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3

Using trigonometry: Real-world applications

Trigonometry is not just theoretical; it has practical applications in various fields such as physics, engineering, and astronomy. Below are some examples of how trigonometric concepts are used in real-life scenarios.

Example 4: Calculating height

To find the height of a building without measuring it directly, the angle of elevation can be measured from a certain distance:

  • Let the distance from the building be d = 50 m.
  • The angle of elevation from this point to the top of the building is θ = 30°.
    • tan(θ) = height / distance Height = Distance × tan(θ) = 50 × tan(30°) = 50 × (1/√3)

    Thus, the height of the building is approximately 28.87 metres.

Through understanding and applying these trigonometric ratios and identities, one can gain deep insight into both mathematical theory and everyday practical problems. Trigonometry, while complex, opens the door to many new ways of thinking about shapes, spaces, and distances.


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