Trigonometry Basic

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It's fundamental to many areas of science, engineering, and even navigation. For competitive exams like the IBPS Clerk, understanding the basics of trigonometry is crucial, especially for solving problems related to heights, distances, and geometric shapes.

What are Trigonometric Ratios?

In a right-angled triangle, the trigonometric ratios relate the angles to the lengths of the sides. Let's consider a right-angled triangle ABC, where angle B is 90 degrees. Let angle C be denoted by θ (theta). The sides opposite to angles A, B, and C are denoted by 'a', 'b', and 'c' respectively.

• The side opposite to the right angle (angle B) is the hypotenuse (h), which is side 'b'.
• The side opposite to angle θ (angle C) is the perpendicular (p), which is side 'c'.
• The side adjacent to angle θ (angle C) is the base (b), which is side 'a'.

The six basic trigonometric ratios are:

1. Sine (sin θ): The ratio of the perpendicular to the hypotenuse.

   sin θ = Perpendicular / Hypotenuse = p / h

2. Cosine (cos θ): The ratio of the base to the hypotenuse.

   cos θ = Base / Hypotenuse = b / h

3. Tangent (tan θ): The ratio of the perpendicular to the base.

   tan θ = Perpendicular / Base = p / b

4. Cosecant (csc θ or cosec θ): The reciprocal of sine.

   csc θ = Hypotenuse / Perpendicular = h / p = 1 / sin θ

5. Secant (sec θ): The reciprocal of cosine.

   sec θ = Hypotenuse / Base = h / b = 1 / cos θ

6. Cotangent (cot θ): The reciprocal of tangent.

   cot θ = Base / Perpendicular = b / p = 1 / tan θ

Fundamental Trigonometric Identities

These are equations that are true for all values of the angles involved. They are derived from the Pythagorean theorem and the definitions of the trigonometric ratios.

1. Pythagorean Identities:
   • sin² θ + cos² θ = 1
   • 1 + tan² θ = sec² θ
   • 1 + cot² θ = csc² θ
2. Reciprocal Identities:
   • sin θ ⋅ csc θ = 1
   • cos θ ⋅ sec θ = 1
   • tan θ ⋅ cot θ = 1
3. Quotient Identities:
   • tan θ = sin θ / cos θ
   • cot θ = cos θ / sin θ

These identities are extremely useful for simplifying trigonometric expressions and solving equations.

Trigonometric Ratios of Standard Angles

There are specific angles for which the values of trigonometric ratios are commonly known and used. These are 0°, 30°, 45°, 60°, and 90°. Memorizing these values is essential for quick problem-solving.

Heights and Distances

Trigonometry is widely used to calculate heights and distances that are difficult or impossible to measure directly. This typically involves right-angled triangles formed by the object, the observer, and the ground.

• Angle of Elevation: The angle formed between the horizontal line from the observer's eye to the object and the line of sight to the object, when the object is above the horizontal line.
• Angle of Depression: The angle formed between the horizontal line from the observer's eye to the object and the line of sight to the object, when the object is below the horizontal line.

In problems involving angles of elevation and depression, we often use tan θ, as it relates the opposite side (height) to the adjacent side (distance).

Example Problem:

A ladder 10 meters long is leaning against a wall. If the ladder makes an angle of 60° with the ground, how high up the wall does the ladder reach?

Solution: Here, the ladder is the hypotenuse (h = 10 m), and the height the ladder reaches on the wall is the perpendicular (p). The angle with the ground is θ = 60°. We use the sine ratio: sin θ = p / h sin 60° = p / 10 We know sin 60° = √3/2. So, √3/2 = p / 10 p = (10 * √3) / 2 p = 5√3 meters. The ladder reaches 5√3 meters up the wall.

Trigonometric Functions for Angles Beyond 90°

Trigonometric functions can be extended to angles greater than 90° using the unit circle. The signs of the trigonometric ratios change depending on the quadrant in which the angle lies.

• Quadrant I (0° to 90°): All trigonometric ratios (sin, cos, tan, csc, sec, cot) are positive.
• Quadrant II (90° to 180°): Only sin and csc are positive.
• Quadrant III (180° to 270°): Only tan and cot are positive.
• Quadrant IV (270° to 360°): Only cos and sec are positive.

Memory Trick for Quadrants: "All Students Take Coffee" A - All positive (Quadrant I) S - Sine positive (Quadrant II) T - Tangent positive (Quadrant III) C - Cosine positive (Quadrant IV)

Trigonometric Identities for Allied Angles

These are relations between trigonometric functions of angles like (90° ± θ), (180° ± θ), (270° ± θ), (360° ± θ).

• If the angle is 90° ± θ or 270° ± θ, the trigonometric ratio changes to its complementary function (sin to cos, cos to sin, tan to cot, cot to tan, sec to csc, csc to sec).
• If the angle is 180° ± θ or 360° ± θ, the trigonometric ratio remains the same.
• The sign of the resulting ratio depends on the quadrant in which the original angle (90° ± θ, etc.) lies.

Examples:

• sin (90° + θ) = cos θ (90°+θ is in Quadrant II, where sin is positive)
• cos (180° - θ) = -cos θ (180°-θ is in Quadrant II, where cos is negative)
• tan (270° + θ) = -cot θ (270°+θ is in Quadrant IV, where tan is negative, and the ratio changes)
• sin (360° - θ) = -sin θ (360°-θ is in Quadrant IV, where sin is negative)

Basic Trigonometric Equations

These are equations involving trigonometric functions that need to be solved for the unknown angle. For competitive exams, typically solutions are sought within a specific range, like 0° to 360°.

Example: Solve sin θ = 1/2 for 0° ≤ θ ≤ 360°. Solution: We know sin 30° = 1/2. This is our principal value. Since sin is positive in Quadrant I and Quadrant II: Quadrant I: θ = 30° Quadrant II: θ = 180° - 30° = 150° So, the solutions are 30° and 150°.

Geometry Basic

Geometry is the study of shapes, sizes, positions of figures, and properties of space. In competitive exams, basic geometry concepts are frequently tested, often integrated with algebra and mensuration. Understanding fundamental shapes, their properties, and basic theorems is key.

Basic Geometric Shapes and Definitions

Points, Lines, and Angles

• Point: A location in space, having no size or dimension.
• Line: A set of points extending infinitely in both directions.
• Line Segment: A part of a line with two endpoints.
• Ray: A part of a line with one endpoint, extending infinitely in one direction.
• Angle: Formed by two rays sharing a common endpoint (vertex). Measured in degrees or radians.

Types of Angles

• Acute Angle: Greater than 0° and less than 90°.
• Right Angle: Exactly 90°.
• Obtuse Angle: Greater than 90° and less than 180°.
• Straight Angle: Exactly 180°.
• Reflex Angle: Greater than 180° and less than 360°.
• Complementary Angles: Two angles whose sum is 90°.
• Supplementary Angles: Two angles whose sum is 180°.
• Adjacent Angles: Angles that share a common vertex and a common side but do not overlap.
• Linear Pair: Two adjacent angles that form a straight line (sum is 180°).
• Vertically Opposite Angles: Angles formed by the intersection of two lines. They are equal.

Parallel and Perpendicular Lines

• Parallel Lines: Lines in a plane that do not intersect, no matter how far they are extended. They are always equidistant.
• Perpendicular Lines: Lines that intersect at a right angle (90°).

When a transversal line intersects two parallel lines:

• Corresponding Angles are equal.
• Alternate Interior Angles are equal.
• Alternate Exterior Angles are equal.
• Consecutive Interior Angles (Interior angles on the same side of the transversal) are supplementary (sum to 180°).

Triangles

A triangle is a polygon with three sides and three angles. The sum of the interior angles of any triangle is always 180°.

Types of Triangles (by Angles):

• Acute-angled Triangle: All three angles are acute (< 90°).
• Right-angled Triangle: One angle is a right angle (90°). The side opposite the right angle is the hypotenuse.
• Obtuse-angled Triangle: One angle is obtuse (> 90°).

Types of Triangles (by Sides):

• Scalene Triangle: All three sides have different lengths, and all three angles are different.
• Isosceles Triangle: Two sides are equal in length, and the angles opposite these sides are equal.
• Equilateral Triangle: All three sides are equal in length, and all three angles are equal (each 60°).

Important Theorems/Properties of Triangles:

• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. (a + b > c, a + c > b, b + c > a)
• Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
• Congruence Rules: Two triangles are congruent if their corresponding sides and angles are equal. Common rules include SSS, SAS, ASA, AAS, RHS (for right-angled triangles).
• Similarity Rules: Two triangles are similar if their corresponding angles are equal and the ratio of their corresponding sides is constant. Common rules include AA, SSS, SAS.

Example: In triangle ABC, if AB = AC, then ∠ABC = ∠ACB.

Area and Perimeter of Triangles:

• Perimeter: Sum of the lengths of all three sides (P = a + b + c).
• Area (Standard Formula): Area = (1/2) * base * height.
• Area (Heron's Formula): If 's' is the semi-perimeter (s = (a+b+c)/2), then Area = √[s(s-a)(s-b)(s-c)].
• Area (Equilateral Triangle): Area = (√3 / 4) * side².
• Area (Right-angled Triangle): Area = (1/2) * product of the two legs (non-hypotenuse sides).

Quadrilaterals

A quadrilateral is a polygon with four sides and four angles. The sum of the interior angles of any quadrilateral is 360°.

Types of Quadrilaterals:

• Parallelogram: Opposite sides are parallel and equal. Opposite angles are equal. Diagonals bisect each other.
• Rectangle: A parallelogram with four right angles. Diagonals are equal and bisect each other.
• Square: A rectangle with all sides equal. Diagonals are equal, bisect each other, and are perpendicular. All angles are 90°.
• Rhombus: A parallelogram with all sides equal. Diagonals bisect each other at right angles. Opposite angles are equal.
• Trapezium (or Trapezoid): At least one pair of opposite sides is parallel.
• Kite: Two pairs of adjacent sides are equal. Diagonals are perpendicular. One diagonal bisects the other.

Area and Perimeter of Quadrilaterals:

• Perimeter: Sum of the lengths of all four sides.
• Area of Parallelogram: Base * height.
• Area of Rectangle: Length * Width.
• Area of Square: Side² or (1/2) * (diagonal)².
• Area of Rhombus: (1/2) * (product of diagonals).
• Area of Trapezium: (1/2) * (sum of parallel sides) * height.

Circles

A circle is a set of points equidistant from a central point.

• Radius (r): The distance from the center to any point on the circle.
• Diameter (d): The distance across the circle through the center (d = 2r).
• Chord: A line segment connecting two points on the circle.
• Circumference (C): The distance around the circle. Formula: C = 2πr or C = πd.
• Area (A): The space enclosed by the circle. Formula: A = πr².
• Sector: A region bounded by two radii and the arc between them.
• Segment: A region bounded by a chord and the arc it subtends.

Value of π (Pi): Approximately 22/7 or 3.14159.

Basic Geometric Theorems

Understanding theorems helps in proving properties and solving complex problems.

• Pythagorean Theorem (for Right-angled Triangles): The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. (hypotenuse² = perpendicular² + base²).
• Thales' Theorem (Intercept Theorem): If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides proportionally.
• Angle in a Semicircle Theorem: The angle subtended by a diameter at any point on the circumference is a right angle (90°).

Coordinate Geometry Basics

Coordinate geometry uses a coordinate system (like the Cartesian plane) to represent geometric shapes.

• Distance Formula: The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²].
• Midpoint Formula: The midpoint of a line segment joining (x₁, y₁) and (x₂, y₂) is [((x₁ + x₂)/2), ((y₁ + y₂)/2)].
• Slope Formula: The slope (m) of a line passing through (x₁, y₁) and (x₂, y₂) is m = (y₂ - y₁) / (x₂ - x₁).

These formulas are crucial for calculating lengths, finding centers of segments, and determining the relationship between lines (parallel, perpendicular).