Trigonometry

This guide summarizes the Trigonometry PDF for competitive exams, making the topic simple with clear explanations, formulas, solved examples, cheatsheets, practice questions (with answers), and visual aids.

1. What is Trigonometry?

Trigonometry is the study of relationships between the sides and angles of triangles, especially right triangles.

• “Tri” = three, “gono” = sides, “metry” = measurement.

2. The Six Trigonometric Ratios

In a right triangle:

Visual: Right Triangle and Ratios

       |\
       | \
   a   |  \  c (hypotenuse)
       |   \
       |____\
         b

sin θ = a/c,   cos θ = b/c,   tan θ = a/b

3. Units and Angle Measurement

• Degree (°): 1 full circle = 360°
• Radian (rad): 1 full circle = 2π radians
• Conversion:

4. Trigonometric Values Table

Note: N.D. = Not Defined

5. Signs of Trigonometric Ratios in Quadrants

• 1st Quadrant: All positive
• 2nd Quadrant: sin, cosec positive
• 3rd Quadrant: tan, cot positive
• 4th Quadrant: cos, sec positive

Visual: ASTC (All Students Take Calculus) Rule

      | 2nd | 1st
   ---+-----+---
      | 3rd | 4th

• Starting from 1st quadrant and going anti-clockwise: All, Sin, Tan, Cos are positive.

6. Key Trigonometric Identities (Cheatsheet)

Pythagorean Identities

• $ \sin^2 θ + \cos^2 θ = 1 $
• $ 1 + \tan^2 θ = \sec^2 θ $
• $ 1 + \cot^2 θ = \csc^2 θ $

Angle Sum and Difference

• $ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B $
• $ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B $
• $ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} $

Double Angle

• $ \sin 2A = 2 \sin A \cos A $
• $ \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A $
• $ \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} $

Half Angle

• $ \sin^2 \frac{A}{2} = \frac{1 - \cos A}{2} $
• $ \cos^2 \frac{A}{2} = \frac{1 + \cos A}{2} $

Product-to-Sum

• $ \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] $
• $ \cos A \cos B = \frac{1}{2} [\cos(A-B) + \cos(A+B)] $
• $ \sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)] $

7. Allied Angles (Special Angle Relationships)

• $ \sin(90^\circ - θ) = \cos θ $
• $ \cos(90^\circ - θ) = \sin θ $
• $ \tan(90^\circ - θ) = \cot θ $
• $ \sin(180^\circ - θ) = \sin θ $
• $ \cos(180^\circ - θ) = -\cos θ $
• $ \tan(180^\circ - θ) = -\tan θ $
• $ \sin(-θ) = -\sin θ $
• $ \cos(-θ) = \cos θ $

8. Laws of Sines and Cosines

• Law of Sines:

• Law of Cosines:

9. Solved Examples

Example 1: Find sin 30°, cos 45°, tan 60°

• $ \sin 30^\circ = \frac{1}{2} $
• $ \cos 45^\circ = \frac{\sqrt{2}}{2} $
• $ \tan 60^\circ = \sqrt{3} $

Example 2: Find a side using trigonometry

Given a right triangle with angle θ = 30°, hypotenuse = 5.

• $ \sin θ = \frac{Opposite}{Hypotenuse} $
• $ \sin 30^\circ = \frac{x}{5} \implies \frac{1}{2} = \frac{x}{5} \implies x = 2.5 $

Example 3: Use Pythagorean Identity

If $ \sin θ = \frac{3}{5} $, find $ \cos θ $.

• $ \sin^2 θ + \cos^2 θ = 1 $
• $ \left(\frac{3}{5}\right)^2 + \cos^2 θ = 1 $
• $ \frac{9}{25} + \cos^2 θ = 1 $
• $ \cos^2 θ = 1 - \frac{9}{25} = \frac{16}{25} $
• $ \cos θ = \pm \frac{4}{5} $

Example 4: Solve tan(x + y) tan(x − y) = 1

• $ \tan(x + y) \tan(x − y) = 1 $
• $ (x + y) + (x − y) = 90^\circ $
• $ 2x = 90^\circ \implies x = 45^\circ $
• $ \tan 45^\circ = 1 $

10. Practice Questions

1. If tan θ = 5/12, find sin θ and cos θ.
2. If sin θ = 3/5, find tan θ and sec θ.
3. If cos θ = 4/5, find sin θ and tan θ.
4. Evaluate: sin 720° – cot 270° – sin 150° cos 120°.
5. If tan(x + y) tan(x − y) = 1, find tan x.
6. If cot 2A cot 3A = 1, find sin 5A cos 5A.
7. If tan θ + cot θ = 2, find tan¹⁰⁰° + cot¹¹⁰°.
8. Solve: tan 1° × tan 2° × tan 3° × … × tan 89°.
9. Solve: cos 1° × cos 2° × … × cos 179°.

11. Visuals & Graphics

A. Unit Circle Diagram

           (0,1)
             |
     (-1,0)-----+-----(1,0)
             |
           (0,-1)

• Shows sine and cosine as y and x coordinates for angle θ.

B. Graphs of Trigonometric Functions

• Sine and Cosine: Wave patterns, period 360° (2π radians).
• Tangent: Repeats every 180° (π radians), vertical asymptotes at 90°, 270°, etc.

C. Quadrant Signs (ASTC)

      II | I
     ----+----
     III | IV

• I: All positive
• II: Sin positive
• III: Tan positive
• IV: Cos positive

12. Quick Reference Table (Cheatsheet)

13. Tips and Tricks

• Memorize values for standard angles (0°, 30°, 45°, 60°, 90°).
• Use ASTC to determine signs in each quadrant.
• For allied angles, reduce to standard angles using identities.
• Use calculator for non-standard angles.

14. Summary

• Trigonometry is essential for geometry, physics, and competitive exams.
• Learn the basic ratios, identities, and standard values.
• Practice with the above examples and questions.
• Use the visuals to remember signs and relationships.

With these explanations, formulas, examples, cheatsheets, practice questions, and visuals, you’re well-equipped to master trigonometry for any exam!