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Class 10 · Chapter 8 · Mathematics

Introduction to Trigonometry Cheat Sheet

Formulae

Trigonometric ratios

sin⁡θ=perpendicularhypotenuse,cos⁡θ=basehypotenuse,tan⁡θ=perpendicularbase\sin\theta = \frac{\text{perpendicular}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{base}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{perpendicular}}{\text{base}}

In a right triangle with angle θ, the side opposite θ is the perpendicular and the side next to θ is the base.

Reciprocal ratios

cosec⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\operatorname{cosec}\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}

These are the reciprocals of sine, cosine and tangent respectively.

Tangent as a ratio

tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}

Valid for values of θ where the denominators are not zero.

Trigonometric ratios of standard angles

sin⁡30∘=12, cos⁡30∘=32, tan⁡45∘=1, sin⁡60∘=32, cos⁡60∘=12\sin 30^\circ = \tfrac{1}{2},\ \cos 30^\circ = \tfrac{\sqrt{3}}{2},\ \tan 45^\circ = 1,\ \sin 60^\circ = \tfrac{\sqrt{3}}{2},\ \cos 60^\circ = \tfrac{1}{2}

The full set for 0°, 30°, 45°, 60° and 90° is used to solve right triangles.

Complementary angle relations

sin⁡(90∘−θ)=cos⁡θ,tan⁡(90∘−θ)=cot⁡θ,sec⁡(90∘−θ)=cosec⁡θ\sin(90^\circ - \theta) = \cos\theta, \quad \tan(90^\circ - \theta) = \cot\theta, \quad \sec(90^\circ - \theta) = \operatorname{cosec}\theta

Two angles are complementary if their sum is 90°.

Theorems

Trigonometric ratios are independent of the triangle size

The values of the trigonometric ratios of an acute angle do not depend on the size of the right triangle in which they are measured; they depend only on the angle.

This follows because right triangles with the same acute angle are similar, so their side ratios are equal.

Identities

First Pythagorean identity

sin⁡2θ+cos⁡2θ=1\sin^{2}\theta + \cos^{2}\theta = 1

Follows from the Pythagoras theorem: perpendicular² + base² = hypotenuse².

Second Pythagorean identity

1+tan⁡2θ=sec⁡2θ1 + \tan^{2}\theta = \sec^{2}\theta

Obtained by dividing sin²θ + cos²θ = 1 throughout by cos²θ.

Third Pythagorean identity

1+cot⁡2θ=cosec⁡2θ1 + \cot^{2}\theta = \operatorname{cosec}^{2}\theta

Obtained by dividing sin²θ + cos²θ = 1 throughout by sin²θ.

Definitions

Trigonometry
The study of relationships between the sides and angles of a triangle.
Trigonometric ratio
A ratio of two sides of a right triangle with respect to one of its acute angles.
Sine (sin θ)
The ratio of the side opposite θ to the hypotenuse.
Cosine (cos θ)
The ratio of the side adjacent to θ (the base) to the hypotenuse.
Tangent (tan θ)
The ratio of the side opposite θ to the side adjacent to θ.
Cosecant, secant and cotangent
The reciprocals of sine, cosine and tangent respectively.
Complementary angles
Two angles whose sum is 90°.

Key Points

  • The trigonometric ratios of an acute angle are fixed for that angle, whatever the size of the right triangle.
  • sin θ and cos θ always lie between 0 and 1; sec θ and cosec θ are never less than 1 in magnitude.
  • tan θ and cot θ can take any real value.
  • The three Pythagorean identities are: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
  • Standard values: sin 0° = 0, sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1.
  • cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = ½, cos 90° = 0.
  • tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3; tan 90° is not defined.
  • Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ and sec(90° − θ) = cosec θ.