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Chapter 8: Introduction to Trigonometry

Trigonometry literally means "triangle measurement". For a right triangle, the angles and sides are related in fixed ways , ratios that depend only on the angle, not on the size of the triangle. These ratios, sin, cos, tan and their reciprocals, are the alphabet of trigonometry.

For boards: 11-mark MCQs on standard angles (0∘,30∘,45∘,60∘,90∘0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ), 22- to 33-mark questions on identities and complementary angles, and 44- to 55-mark proofs of identities. The next chapter applies these tools to heights and distances; together they account for a sizeable chunk of the paper.

Beyond exams, trigonometry powers waves (physics), surveying, navigation (latitude/longitude), computer graphics, music synthesis, and signal processing. Every periodic phenomenon in the universe is eventually written down in terms of sin and cos.

What's inside

  • Defining the six trigonometric ratios , sin, cos, tan, cosec, sec, cot.
  • Standard angles , 0,30,45,60,90∘0, 30, 45, 60, 90^\circ.
  • Trigonometric identities , sin⁡2+cos⁡2=1\sin^2 + \cos^2 = 1, 1+tan⁡2=sec⁡21 + \tan^2 = \sec^2, 1+cot⁡2=csc⁡21 + \cot^2 = \csc^2.
  • Complementary angles , sin⁡(90−θ)=cos⁡θ\sin(90 - \theta) = \cos \theta and friends.
  • Proving identities , strategies and examples.

Key results / Formula card

RatioDefinitionReciprocal
sin⁡θ\sin \thetaopposite / hypotenusecsc⁡θ\csc \theta
cos⁡θ\cos \thetaadjacent / hypotenusesec⁡θ\sec \theta
tan⁡θ\tan \thetaopposite / adjacentcot⁡θ\cot \theta
IdentityForm
Pythagoreansin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Tangent–secant1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
Cotangent–cosecant1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta
Quotienttan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}, \quad \cot\theta = \dfrac{\cos\theta}{\sin\theta}
Anglesin⁡\sincos⁡\costan⁡\tan
0∘0^\circ001100
30∘30^\circ12\tfrac{1}{2}32\tfrac{\sqrt{3}}{2}13\tfrac{1}{\sqrt{3}}
45∘45^\circ12\tfrac{1}{\sqrt{2}}12\tfrac{1}{\sqrt{2}}11
60∘60^\circ32\tfrac{\sqrt{3}}{2}12\tfrac{1}{2}3\sqrt{3}
90∘90^\circ1100undefined
ComplementaryIdentity
Sine and cosinesin⁡(90∘−θ)=cos⁡θ\sin(90^\circ - \theta) = \cos\theta
Cosine and sinecos⁡(90∘−θ)=sin⁡θ\cos(90^\circ - \theta) = \sin\theta
Tan and cottan⁡(90∘−θ)=cot⁡θ\tan(90^\circ - \theta) = \cot\theta

Sub-topics

5 pages
01
Trigonometric ratios
Define sin, cos, tan, cosec, sec and cot from a right triangle and learn how to read each ratio off a labelled diagram.
02
Trigonometric ratios of standard angles
Compute and memorise sin, cos, tan of 0, 30, 45, 60, 90 degrees from standard right triangles.
03
Trigonometric identities
The three Pythagorean identities, derivation from a right triangle, and strategies for proving harder identities.
04
Complementary angles
Identities like sin(90 - theta) = cos theta and their uses in simplification.
05
Proving trigonometric identities , practice
Strategies and worked examples for the trickier identities that show up on board exams.

Practice quiz

Answer the questions; explanations appear after each.

Quiz
Chapter 8 : Mixed practice
10 questions · pick the best answer
Q1

sin⁡30°\sin 30° equals:

Q2

If sin⁡θ=3/5\sin\theta = 3/5, then cos⁡θ\cos\theta (acute) is:

Q3

sin⁡230°+cos⁡230°\sin^2 30° + \cos^2 30°:

Q4

tan⁡45°⋅cot⁡45°\tan 45° \cdot \cot 45°:

Q5

sin⁡(90°−θ)=\sin(90° - \theta) = :

Q6

If tan⁡θ=1\tan\theta = 1, then θ=\theta = :

Q7

sec⁡2θ−tan⁡2θ\sec^2\theta - \tan^2\theta:

Q8

sin⁡60°⋅cos⁡30°\sin 60° \cdot \cos 30°:

Q9

If sin⁡3A=cos⁡(A−26°)\sin 3A = \cos(A - 26°), then AA:

Q10

sin⁡65°/cos⁡25°\sin 65° / \cos 25°: