CBSE · Class 10 · Mathematics · Chapter 9

Some Applications of Trigonometry — Formula Sheet

Board Formulas

12 formulas

1.Angle of Elevation★
$tan θ = \frac{h}{d}$
$θ$ : Angle of elevation (degrees) · $h$ : Height of the object above eye level (m) · $d$ : Horizontal distance from the observer to the foot of the object (m)
The angle between the line of sight and the horizontal when the object is ABOVE the eye. The line of sight is the line from the observer's eye to the object.
2.Height and Distance from tan
$h = d tan θ, d = \frac{h}{t a n θ} = h cot θ$
Rearranged forms. Find h when the distance is known; find d when the height is known.
3.Slant Length: sin and cos
$h = l sin θ, d = l cos θ$
$l$ : Slant length — hypotenuse of the right triangle (m) · $h$ : Vertical height (m) · $d$ : Horizontal distance (m)
l is the hypotenuse: the length of a kite string, a ladder, a rope or the line of sight itself. θ is measured from the horizontal.
4.Angle of Depression★
$θ_{depression} = θ_{elevation} (alternate angles)$
The angle between the line of sight and the horizontal when the object is BELOW the eye. The depression of B seen from A equals the elevation of A seen from B, because the two horizontals are parallel.
5.Allowing for the Observer's Height★
$H = h_{o} + d tan θ$
$H$ : Total height of the object (m) · $h_{o}$ : Height of the observer's eyes above the ground (m) · $d$ : Horizontal distance from the observer to the object (m)
If the angle is measured from the observer's eyes, d tan θ gives only the part ABOVE eye level. Add the eye height.
6.tan Values Used in This Chapter
$tan 3 0^{\circ} = \frac{1}{3}, tan 4 5^{\circ} = 1, tan 6 0^{\circ} = 3$
cot is the reciprocal: cot 30° = √3, cot 45° = 1, cot 60° = 1/√3.
7.sin and cos Values Used in This Chapter
$sin 3 0^{\circ} = cos 6 0^{\circ} = \frac{1}{2}, sin 6 0^{\circ} = cos 3 0^{\circ} = \frac{3}{2}, sin 4 5^{\circ} = cos 4 5^{\circ} = \frac{1}{2}$
Needed for string, ladder and slant-distance problems.
8.Shadow Length
$s = h cot θ = \frac{h}{t a n θ}$
$s$ : Length of the shadow (m) · $h$ : Height of the vertical object (m)
θ is the sun's altitude (angle of elevation of the sun). The higher the sun, the shorter the shadow.
9.Two Points on the Same Side★
$x = h (cot α - cot β) \Rightarrow h = \frac{x}{c o t α - c o t β}$
$x$ : Distance between the two observation points (m) · $h$ : Height of the object (m) · $α, β$ : Angles of elevation from the farther and nearer points
Angles α (farther point) and β (nearer point), α < β, observed at two points in line with the foot. Also gives the distance between two ships seen from a lighthouse, or the change in a shadow's length.
10.Special Case: 30° and 60°
$α = 3 0^{\circ}, β = 6 0^{\circ} \Rightarrow h = \frac{3}{2} x, B D = \frac{x}{2}$
A handy check: if the elevation changes from 30° to 60° after walking x towards the tower, the tower is (√3/2)x high and you are now x/2 from its foot (BD = distance of the nearer point from the foot B). Show the full working in the exam.
11.Two Points on Opposite Sides
$x = h (cot α + cot β)$
For two points on opposite sides of the object's foot (or two poles of equal height h seen from a point between them). x is the total distance between the two points.
12.Building and Tower
$d = h cot β, H = h + d tan α$
$h$ : Height of the building (m) · $H$ : Height of the tower (m) · $d$ : Horizontal distance between building and tower (m)
From the top of a building of height h: β is the angle of depression of the tower's foot, α is the angle of elevation of the tower's top. The horizontal through the building's top splits the tower into h and d tan α.