Maharashtra State Board · Class 10 · Mathematics · Geometry Chapter 6

Trigonometry — Formula Sheet

Board Formulas

10 formulas

1.Basic Trigonometric Ratios (SOH-CAH-TOA)★
$sin θ = \frac{Opp}{Hyp}, cos θ = \frac{Adj}{Hyp}, tan θ = \frac{Opp}{Adj}$
$θ$ : Angle of interest (0° < θ < 90°) · $O pp$ : Side opposite to θ · $A d j$ : Side adjacent to θ (not hypotenuse) · $H y p$ : Hypotenuse — side opposite the right angle
Only for a right-angled triangle. Ratios depend on the angle, not on triangle size.
2.Reciprocal Ratios
$csc θ = \frac{1}{s i n θ}, sec θ = \frac{1}{c o s θ}, cot θ = \frac{1}{t a n θ}$
Cosec, sec, cot are reciprocals of sin, cos, tan respectively.
3.Quotient Identity
$tan θ = \frac{s i n θ}{c o s θ}, cot θ = \frac{c o s θ}{s i n θ}$
Rewrite tan and cot in terms of sin and cos.
4.Pythagorean Identity 1★
$sin^{2} θ + cos^{2} θ = 1$
The fundamental identity. Rearranges to sin²θ = 1 − cos²θ or cos²θ = 1 − sin²θ.
5.Pythagorean Identity 2
$1 + tan^{2} θ = sec^{2} θ$
Divide identity 1 by cos²θ.
6.Pythagorean Identity 3
$1 + cot^{2} θ = csc^{2} θ$
Divide identity 1 by sin²θ.
7.Complementary Angle Relations★
$sin (9 0^{\circ} - θ) = cos θ, cos (9 0^{\circ} - θ) = sin θ$
Ratio of θ = co-ratio of (90° − θ). Also tan(90° − θ) = cot θ.
8.Standard Angle Values
$sin 3 0^{\circ} = \frac{1}{2}, sin 4 5^{\circ} = \frac{1}{2}, sin 6 0^{\circ} = \frac{3}{2}$
For cos, read the same values in reverse order. tan = sin/cos.
9.Angle of Elevation
$tan (θ_{e}) = \frac{height}{horizontal distance}$
Measured upward from the horizontal to the line of sight to the object.
10.Angle of Depression
$tan (θ_{d}) = \frac{height}{horizontal distance}$
Measured downward from the horizontal. In a right triangle it equals the alternate angle of elevation from the object.