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 ↓


Basic Trigonometric Ratios


Trigonometry is the study of the relationships between the angles and sides of triangles. In mathematics, trigonometric ratios are the ratios of the lengths of the sides of a right triangle with respect to its angles. The three basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).

Understanding right-angle triangles

A right triangle is a type of triangle with one angle equal to 90 degrees. The side opposite this right angle is called the hypotenuse, which is the longest side of the triangle. The other two sides are called the opposite side and the adjacent side, each named in relation to the given angle. Here's a quick illustration:

       ,
       ,
    o |  h
       ,
       ,
         A

In the above triangle, where:

  • h is the hypotenuse
  • o is the side that is opposite to the angle we are referring to
  • a is the side adjacent to the angle we are referring to

Basic trigonometric ratios

The trigonometric ratios for any angle of a right-angled triangle are defined as follows:

Chord

The sine (θ) of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

    sin(θ) = opposite/hypotenuse

Cosine

The cosine (θ) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

    cos(θ) = adjacent / hypotenuse

Tangent line

The tangent (θ) of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

    tan(θ) = opposite/adjacent

Visual example

Let's explain it with a visual representation, where we have a right triangle and an angle θ:

θ near Antonym Ear

Example calculation

Example 1: Calculating Sine

Suppose we have a right-angled triangle with angle θ, in which the opposite side is 3 units and the hypotenuse is 5 units.

    sin(θ) = opposite / hypotenuse = 3/5 = 0.6

Therefore, sin(θ) = 0.6.

Example 2: Calculating Cosine

Suppose the adjacent side of the same triangle is 4 units and the hypotenuse is 5 units.

    cos(θ) = adjacent / hypotenuse = 4/5 = 0.8

Thus, cos(θ) = 0.8.

Example 3: Calculating the Tangent

Calculate the tangent using the same properties of a triangle.

    tan(θ) = opposite/adjacent = 3/4 = 0.75

So, tan(θ) = 0.75.

Applications of trigonometric ratios

Trigonometric ratios have various applications in different fields such as engineering, physics and even in simple daily tasks like architecture and navigation. Understanding these basic trigonometric concepts allows us to solve problems involving right-angled triangles and perform important calculations in designs and structures.

Practice exercise

Let's further strengthen our understanding by doing some exercises. Calculate the trigonometric ratios for the following triangle:

A right-angled triangle is given:

  • Opposite side = 7 units
  • Adjacent side = 24 units
  • Hypotenuse = 25 units

Calculate sin(θ), cos(θ), and tan(θ)

Solution:

Sign:

        sin(θ) = 7 / 25 = 0.28

Cosine:

        cos(θ) = 24 / 25 = 0.96

Tangent line:

        tan(θ) = 7 / 24 = 0.2917

Conclusion

The basic trigonometric ratios of sine, cosine, and tangent are fundamental elements in trigonometry. They help us understand and solve problems involving triangles. Mastering these ratios is important as they are used extensively in more advanced mathematics and various real-world applications.


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