CBSE · Class 10 · Mathematics · Chapter 8

Introduction to Trigonometry — Formula Sheet

Board Formulas

10 formulas

1.Sine of an Angle★
$sin θ = \frac{opposite}{hypotenuse}$
In a right triangle, opposite to angle θ over the hypotenuse. Value between 0 and 1 for 0° < θ < 90°.
2.Cosine of an Angle
$cos θ = \frac{adjacent}{hypotenuse}$
Adjacent (next to θ, not the hypotenuse) over hypotenuse.
3.Tangent of an Angle★
$tan θ = \frac{opposite}{adjacent} = \frac{s i n θ}{c o s θ}$
Sine over cosine. Undefined when cos θ = 0 (θ = 90°).
4.Reciprocal Ratios
$cosec θ = \frac{1}{s i n θ}, sec θ = \frac{1}{c o s θ}, cot θ = \frac{1}{t a n θ} = \frac{c o s θ}{s i n θ}$
cosec, sec, cot are just reciprocals. Learn all six together.
5.Pythagorean Identity (Sine-Cosine)★
$sin^{2} θ + cos^{2} θ = 1$
The most-used identity. Follows directly from Pythagoras.
6.Identity: 1 + tan²
$1 + tan^{2} θ = sec^{2} θ$
Divide the Pythagorean identity by cos²θ.
7.Identity: 1 + cot²
$1 + cot^{2} θ = cosec^{2} θ$
Divide the Pythagorean identity by sin²θ.
8.Complementary Angle Identities
$sin (9 0^{\circ} - θ) = cos θ, tan (9 0^{\circ} - θ) = cot θ, sec (9 0^{\circ} - θ) = cosec θ$
'Co' functions swap under 90° − θ. Symmetry of a right triangle explains this.
9.Standard Angle Values (Sine)★
$sin 0^{\circ} = 0, sin 3 0^{\circ} = \frac{1}{2}, sin 4 5^{\circ} = \frac{1}{2}, sin 6 0^{\circ} = \frac{3}{2}, sin 9 0^{\circ} = 1$
Memorise this row; cos is the reverse (cos 0° = 1, cos 90° = 0).
10.Standard Angle Values (Tangent)
$tan 0^{\circ} = 0, tan 3 0^{\circ} = \frac{1}{3}, tan 4 5^{\circ} = 1, tan 6 0^{\circ} = 3, tan 9 0^{\circ} = undefined$
tan = sin/cos; the 90° value is undefined because cos 90° = 0.