Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is particularly useful in dealing with problems involving distances, heights, and angles, which are common in various engineering, physics, and navigation applications. The fundamental concepts revolve around trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. These functions are defined for angles and relate them to the ratios of the sides of a right-angled triangle.

Right-Angled Triangle Fundamentals

Consider a right-angled triangle ABC, where angle B is 90 degrees. Let angle C be denoted by θ. The side opposite to the right angle (AC) is called the Hypotenuse (H). The side opposite to angle θ (AB) is called the Perpendicular (P) or Opposite. The side adjacent to angle θ (BC) is called the Base (B) or Adjacent.

The six trigonometric ratios are defined as follows:

  • Sine (sin θ) = Opposite / Hypotenuse = P/H
  • Cosine (cos θ) = Base / Hypotenuse = B/H
  • Tangent (tan θ) = Opposite / Base = P/B
  • Cosecant (cosec θ or csc θ) = 1 / sin θ = H/P
  • Secant (sec θ) = 1 / cos θ = H/B
  • Cotangent (cot θ) = 1 / tan θ = B/P

It's important to remember these definitions. A simple mnemonic to remember the first three ratios is SOH CAH TOA:

SOH CAH TOA:
  • Sin = Opposite / Hypotenuse
  • Cos = Adjacent / Hypotenuse
  • Tan = Opposite / Adjacent

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. They are fundamental tools for simplifying trigonometric expressions and solving trigonometric equations.

Fundamental Identities

These are the most basic and widely used identities.

  1. Pythagorean Identity:

    This identity is derived from the Pythagorean theorem (a² + b² = c²) applied to a right-angled triangle. sin2 θ + cos2 θ = 1

    From this, we can derive: sin2 θ = 1 - cos2 θ cos2 θ = 1 - sin2 θ

  2. Tangent Identity:

    tan θ = sin θ / cos θ

  3. Cotangent Identity:

    cot θ = cos θ / sin θ

Other Important Identities

These are derived from the fundamental Pythagorean identity by dividing by sin2 θ or cos2 θ.

  1. 1 + tan2 θ = sec2 θ

    This can be rewritten as: sec2 θ - tan2 θ = 1 tan2 θ = sec2 θ - 1

  2. 1 + cot2 θ = cosec2 θ

    This can be rewritten as: cosec2 θ - cot2 θ = 1 cot2 θ = cosec2 θ - 1

Memory Trick for Identities:

Imagine a table with three rows. The first row has sin, cos, tan. The second row has cosec, sec, cot. The third row has 1, 1, 1. For the Pythagorean identities, think of the row number. Row 1: sin2 θ + cos2 θ = 1 Row 2: 1 + tan2 θ = sec2 θ (This is like Row 1 shifted, with 1 added to the left) Row 3: 1 + cot2 θ = cosec2 θ (This is like Row 2 shifted)

Trigonometric Ratios of Standard Angles

Certain angles have specific, commonly used trigonometric values that are essential to memorize for quick problem-solving. These are typically 0°, 30°, 45°, 60°, and 90°.

Angle (θ) sin θ cos θ tan θ cosec θ sec θ cot θ
0° 0 1 0 Undefined 1 Undefined
30° 1/2 √3/2 1/√3 2 2/√3 √3
45° 1/√2 1/√2 1 √2 √2 1
60° √3/2 1/2 √3 2/√3 2 1/√3
90° 1 0 Undefined 1 Undefined 0
Shortcut for Standard Angle Values:

For sin values: Write 0, 1, 2, 3, 4. Take the square root of each: √0, √1, √2, √3, √4 which is 0, 1, √2, √3, 2. Divide each by 2: 0/2, 1/2, √2/2, √3/2, 2/2. This gives you sin 0°, 30°, 45°, 60°, 90° as: 0, 1/2, 1/√2, √3/2, 1. For cos values: Simply write the sin values in reverse order. For tan values: tan θ = sin θ / cos θ. Calculate each value using the sin and cos values. For cosec, sec, cot: These are the reciprocals of sin, cos, tan respectively.

Trigonometric Ratios of Allied Angles

Allied angles are angles that are related to a reference angle (usually 90°, 180°, 270°, 360°) by addition or subtraction. Understanding their trigonometric ratios is crucial. The signs of the trigonometric functions depend on the quadrant in which the angle lies.

Quadrants and Signs

The coordinate plane is divided into four quadrants by the x and y axes.

  • Quadrant I (0° to 90°): All trigonometric ratios (sin, cos, tan, etc.) are positive.
  • Quadrant II (90° to 180°): Only sin and cosec are positive.
  • Quadrant III (180° to 270°): Only tan and cot are positive.
  • Quadrant IV (270° to 360°): Only cos and sec are positive.
Mnemonic for Quadrant Signs: "All Students Take Coffee"
  • All (Quadrant I) - All positive
  • Students (Quadrant II) - Sine positive
  • Take (Quadrant III) - Tangent positive
  • Coffee (Quadrant IV) - Cosine positive

Angle Transformations

When dealing with angles like (90° ± θ), (180° ± θ), (270° ± θ), and (360° ± θ), the trigonometric function may change (from sin to cos, or tan to cot, etc.) or stay the same. The sign is determined by the quadrant.

  • If the angle is a multiple of 180° (e.g., 180° ± θ, 360° ± θ): The trigonometric function remains the same. (sin remains sin, cos remains cos, etc.)
  • If the angle is a multiple of 90° but not 180° (e.g., 90° ± θ, 270° ± θ): The trigonometric function changes to its co-function. (sin changes to cos, cos to sin, tan to cot, cot to tan, sec to cosec, cosec to sec).

Examples:

  • sin (180° - θ) = sin θ (180° - θ is in Quadrant II, where sin is positive)
  • cos (180° + θ) = -cos θ (180° + θ is in Quadrant III, where cos is negative)
  • tan (90° + θ) = -cot θ (90° + θ is in Quadrant II, where tan is negative, and 90° causes the change to cot)
  • sin (270° - θ) = -cos θ (270° - θ is in Quadrant III, where sin is negative, and 270° causes the change to cos)
  • cos (360° - θ) = cos θ (360° - θ is in Quadrant IV, where cos is positive)
  • tan (360° + θ) = tan θ (360° + θ is in Quadrant I, where tan is positive)

Trigonometric Ratios of Negative Angles

The trigonometric ratios of negative angles can be expressed in terms of positive angles.

  • sin (-θ) = -sin θ (Odd function)
  • cos (-θ) = cos θ (Even function)
  • tan (-θ) = -tan θ (Odd function)
  • cosec (-θ) = -cosec θ
  • sec (-θ) = sec θ
  • cot (-θ) = -cot θ

Notice that only cosine and secant are even functions (their value doesn't change for a negative angle).

Trigonometric Ratios of Complementary Angles

Two angles are complementary if their sum is 90°.

  • sin (90° - θ) = cos θ
  • cos (90° - θ) = sin θ
  • tan (90° - θ) = cot θ
  • cot (90° - θ) = tan θ
  • sec (90° - θ) = cosec θ
  • cosec (90° - θ) = sec θ

Example: sin 30° = cos (90° - 30°) = cos 60°. Both are 1/2.

Trigonometric Ratios of Supplementary Angles

Two angles are supplementary if their sum is 180°.

  • sin (180° - θ) = sin θ
  • cos (180° - θ) = -cos θ
  • tan (180° - θ) = -tan θ
  • cot (180° - θ) = -cot θ
  • sec (180° - θ) = -sec θ
  • cosec (180° - θ) = cosec θ

Notice that only sine and cosecant are positive for supplementary angles.

Height and Distance Problems

Trigonometry is extensively used to solve problems involving heights and distances without actually measuring them directly. This typically involves forming right-angled triangles using the given information.

Key Terms

  • 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.

Solving Problems

  1. Draw a Diagram: Sketch the situation described in the problem. Label the known quantities (heights, distances, angles) and the unknown quantity you need to find.
  2. Identify Right-Angled Triangles: The diagram will usually form one or more right-angled triangles.
  3. Choose the Correct Trigonometric Ratio: Based on the angle you know and the sides you need (opposite, adjacent, hypotenuse), select the appropriate trigonometric ratio (SOH CAH TOA).
  4. Set up an Equation: Write an equation using the chosen ratio and the known/unknown values.
  5. Solve the Equation: Solve the equation for the unknown quantity.
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:

  • The ladder is the hypotenuse (H = 10m).
  • The height up the wall is the side opposite to the 60° angle (P).
  • We need a ratio involving Opposite and Hypotenuse, which is Sine.
  • sin 60° = Opposite / Hypotenuse
  • √3/2 = P / 10
  • P = 10 * (√3/2)
  • P = 5√3 meters
The ladder reaches 5√3 meters up the wall.

Trigonometric Equations

Trigonometric equations involve trigonometric functions of unknown variables. Solving them means finding the values of the variable for which the equation holds true.

General Solutions

Trigonometric functions are periodic, meaning they repeat their values. Therefore, a single trigonometric equation can have infinitely many solutions. The general solution expresses all possible solutions in a concise form.

Here are some standard general solution formulas:

  • If sin x = sin α, then x = nπ + (-1)n α, where n is an integer. (In degrees: x = n * 180° + (-1)n α)
  • If cos x = cos α, then x = 2nπ ± α, where n is an integer. (In degrees: x = n * 360° ± α)
  • If tan x = tan α, then x = nπ + α, where n is an integer. (In degrees: x = n * 180° + α)

Note: α is the principal value (usually the smallest positive angle) that satisfies the equation.

Key Point for Exam:

Memorize the standard values of trigonometric ratios for angles 0°, 30°, 45°, 60°, 90° and the fundamental identities. Practice problems involving heights and distances, as they are frequently asked. Understanding the quadrant rules for signs and co-function changes is also critical.

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