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 Ratios


Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It plays an important role in many aspects of mathematics, physics, engineering and various fields where the measurement of angles and distances is necessary. One of the most fundamental concepts in trigonometry is the trigonometric ratio.

Understanding trigonometric ratios

The three sides in a right-angled triangle are:

  • Hypotenuse: The side opposite the right angle, and it is the longest side of the triangle.
  • Opposite side (of an angle): The side that is opposite to the angle we are considering.
  • Adjacent side (of an angle): The side next to the angle we are considering, excluding the hypotenuse.

Six trigonometric ratios

The main trigonometric ratios are defined with respect to the acute angles in a right triangle. The six trigonometric ratios are:

1. Sin (sin)

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

sin θ = Opposite / Hypotenuse

For example, consider a right-angled triangle with one of the angles marked θ:

near Antonym Ear

If the opposite side is 3 units and the hypotenuse is 5 units:

sin θ = 3 / 5 = 0.6

2. Cosine (cos)

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

cos θ = Adjacent / Hypotenuse

In the same triangle example:

cos θ = 4 / 5 = 0.8

3. Tangent (Tan)

The 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

Continuing your example:

tan θ = 3 / 4 = 0.75

Reciprocal trigonometric ratios

There are also three reciprocal trigonometric ratios:

4. Cosecant (CSC)

The cosecant of an angle is the reciprocal of the sine. It is the ratio of the hypotenuse and the opposite side.

csc θ = 1/sin θ = Hypotenuse / Opposite

So in our example:

csc θ = 5 / 3 ≈ 1.67

5. Second (sec)

The secant of an angle is the inverse of the angle cosine. It is the ratio of the hypotenuse and the adjacent side.

sec θ = 1/cos θ = Hypotenuse / Adjacent

From our example:

sec θ = 5 / 4 = 1.25

6. Cotangent (cot)

The cotangent of an angle is the reciprocal of the tangent. It is the ratio of the adjacent side to the opposite side.

cot θ = 1/tan θ = Adjacent / Opposite

Use our example:

cot θ = 4 / 3 ≈ 1.33

Mnemonic for remembering trigonometric ratios

A popular mnemonic for remembering trigonometric ratios is "SOH-CAH-TOA":

  • SOH: Sine = opposite / hypotenuse
  • CAH: Cosine = adjacent / hypotenuse
  • TOA: tangent = opposite/adjacent

Example problems

Example 1:

A ladder rests against a wall. The bottom of the ladder is 4 m from the wall, and it reaches a height of 3 m on the wall. What angle does the ladder make with the ground?

Solution:

Here, we can use the tangent ratio:

tan θ = Opposite / Adjacent = 3 / 4

Use a calculator to find θ ≈ tan -1 (0.75) ≈ 36.87°.

Example 2:

The hypotenuse of a right-angled triangle is 13 units, and one of its angles is 30°. Find the length of the opposite and adjacent sides.

Solution:

  • Using sin(30°):
    sin(30°) = Opposite / 13 = 0.5

    Opposite = 0.5 × 13 = 6.5

  • Using cos(30°):
    cos(30°) = Adjacent / 13 ≈ 0.866

    adjacent = 0.866 × 13 ≈ 11.26

Understanding with more visuals

Consider the following representation of a right-angled triangle with angle θ:

θ near Antonym Ear

For θ, trigonometric ratios are often used to express three major side relationships. Sine, cosine, and tangent ratios are fundamentally important for solving various problems in trigonometry.

Applications of trigonometric ratios

Trigonometric ratios have numerous applications, ranging from practical tasks in everyday activities to solving complex problems in various fields:

  • Surveying: Trigonometry is used in surveying to measure distances between distant objects. Distances combined with angles help in determining various parameters.
  • Navigation: Trigonometry helps navigate around curves and angles. It is used in route planning, shipbuilding and aviation industries to determine paths and find positions.
  • Architecture and Engineering: It helps in designing buildings, bridges and other structures, determining heights, forces and loads.

Conclusion

Trigonometric ratios are a key component of trigonometry. By using the relationships between the angles and side lengths of right triangles, these ratios help solve many practical problems. An understanding of sine, cosine, tangent, and their inverses is foundational to further study in math and science.


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Trigonometric Ratios

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