45  Trigonometry

45.1 Introduction

Trigonometry, derived from the Greek trigonon (triangle) and metron (measure), studies the relationships between angles and sides of triangles.
Its scope extends far beyond geometry: it is crucial in physics, astronomy, engineering, navigation, and competitive exams.
For aptitude exams, the focus is on ratios, identities, transformations, equations, and applications like heights and distances.

45.2 Measurement of Angles

  • Degree: \(360^\circ\) in a full circle, \(90^\circ\) in a right angle.
  • Radian: Central angle subtended by arc equal to radius.
  • Relation: \(180^\circ = \pi\) radians.

Conversions:
- Degrees → radians: multiply by \(\pi/180\).
- Radians → degrees: multiply by \(180/\pi\).

Examples:
- \(60^\circ = \pi/3\) radians.
- \(3\pi/4\) rad = \(135^\circ\).

45.3 Trigonometric Ratios

For right \(\triangle ABC\) with \(\angle A=\theta\): - \(\sin \theta = \tfrac{\text{opposite}}{\text{hypotenuse}}\)
- \(\cos \theta = \tfrac{\text{adjacent}}{\text{hypotenuse}}\)
- \(\tan \theta = \tfrac{\text{opposite}}{\text{adjacent}}\)
- \(\cot \theta = \tfrac{1}{\tan \theta}\)
- \(\sec \theta = \tfrac{1}{\cos \theta}\)
- \(\csc \theta = \tfrac{1}{\sin \theta}\)

45.4 Trigonometric Values of Standard Angles

\(\theta\) \(0^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(90^\circ\)
\(\sin \theta\) 0 1/2 \(\tfrac{\sqrt{2}}{2}\) \(\tfrac{\sqrt{3}}{2}\) 1
\(\cos \theta\) 1 \(\tfrac{\sqrt{3}}{2}\) \(\tfrac{\sqrt{2}}{2}\) 1/2 0
\(\tan \theta\) 0 \(1/\sqrt{3}\) 1 \(\sqrt{3}\) ∞

Also know values for \(120^\circ, 135^\circ, 150^\circ, 180^\circ\), etc.

45.5 Signs in Quadrants

Using “All Students Take Care” mnemonic:
- Quadrant I: all positive.
- Quadrant II: \(\sin,\csc\) positive.
- Quadrant III: \(\tan,\cot\) positive.
- Quadrant IV: \(\cos,\sec\) positive.

45.6 Fundamental Identities

  1. \(\sin^2\theta+\cos^2\theta=1\)
  2. \(1+\tan^2\theta=\sec^2\theta\)
  3. \(1+\cot^2\theta=\csc^2\theta\)

45.7 Co-function Relationships

  • \(\sin(90^\circ-\theta)=\cos \theta\)
  • \(\cos(90^\circ-\theta)=\sin \theta\)
  • \(\tan(90^\circ-\theta)=\cot \theta\)
  • \(\csc(90^\circ-\theta)=\sec \theta\)
  • \(\sec(90^\circ-\theta)=\csc \theta\)
  • \(\cot(90^\circ-\theta)=\tan \theta\)

45.8 Compound Angles

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

45.9 Multiple and Sub-multiple Angles

  • \(\sin 2A=2\sin A\cos A\)
  • \(\cos 2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A\)
  • \(\tan 2A=\tfrac{2\tan A}{1-\tan^2A}\)

Half-angle:
- \(\sin^2(A/2)=\tfrac{1-\cos A}{2}\)
- \(\cos^2(A/2)=\tfrac{1+\cos A}{2}\)

45.10 Product-to-Sum and Sum-to-Product

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

45.11 Trigonometric Equations

Solve using identities and periodicity.

Examples:
- \(\sin x=0 \Rightarrow x=n\pi\).
- \(\cos x=0 \Rightarrow x=\pi/2+n\pi\).
- \(\tan x=0 \Rightarrow x=n\pi\).

45.12 Graphs of Trigonometric Functions

  • Sine: wave from \(-1\) to \(1\), period \(2\pi\).
  • Cosine: same shape shifted left.
  • Tangent: repeats every \(\pi\), with asymptotes.

Understanding graphs helps in inequalities and solving equations.

45.13 Heights and Distances

Trigonometry is used to calculate distances/heights indirectly.
- Angle of elevation: upwards from horizontal.
- Angle of depression: downwards from horizontal.

Example: A tower casts a shadow of 20 m when Sun’s elevation is \(30^\circ\).
Height = \(20\tan 30^\circ=20/\sqrt{3}\).

Types of problems: - Single tower, single angle.
- Single tower, two angles.
- Two objects (boats, poles, airplanes).
- Angles of elevation and depression combined.

45.14 Solved Examples

Example 1
If \(\sin A=3/5\), find \(\cos A\).
\(\cos A=\sqrt{1-\sin^2 A}=\sqrt{1-9/25}=4/5\).

Example 2
Solve \(\sin^2 x-\cos^2 x=0\).
\(\tan^2 x=1 \Rightarrow x=45^\circ,135^\circ,\dots\).

Example 3
From a point 48 m from the base of a tower, angle of elevation is \(30^\circ\).
Height = \(48\tan 30^\circ=16\sqrt{3}\) m.

45.15 Practice Set – Level 1

  1. Convert \(225^\circ\) into radians.
  2. Find \(\sin 45^\circ\cos 45^\circ\).
  3. If \(\tan \theta=5/12\), find \(\sin \theta\) and \(\cos \theta\).
  4. Simplify \(\frac{1-\cos 2A}{\sin^2 A}\).
  5. A tower casts 100 m shadow when Sun’s altitude is \(45^\circ\). Find height.

45.16 Practice Set – Level 2

  1. Solve \(\cos x=\sin x\).
  2. Find general solution of \(\tan^2 x=1\).
  3. Prove: \(\frac{1-\tan^2 A}{1+\tan^2 A}=\cos 2A\).
  4. A building subtends angles of elevation \(30^\circ\) and \(60^\circ\) at two points 50 m apart. Find height.
  5. Find value of \(\sin^2 18^\circ+\sin^2 72^\circ\).

45.17 Summary

  • Trigonometry unites angles and ratios, essential for aptitude.
  • Must know: identities, quadrant signs, compound/multiple angle formulas.
  • Applications: heights, distances, navigation, oscillations.
  • Solve with multipliers, diagrams, and identities for speed.
  • Practice with both basic and advanced problems for exams.