Some Applications of Trigonometry – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Some Applications of Trigonometry chapter of Mathematic. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Height of an Object: h = d × tan(θ)
h is the height of the object, d is the distance from the object, and θ is the angle of elevation. This formula is used to determine the height of an inaccessible object by measuring the angle from a known distance.
Distance from Object: d = h / tan(θ)
d is the distance to the object, h is the height of the object, and θ is the angle of elevation. Used to calculate how far one should stand to measure an object's height.
Length of Ladder: L = h / sin(θ)
L is the length of the ladder, h is the vertical height to be reached, and θ is the angle of elevation of the ladder to the horizontal ground. This is used for determining ladder length needed to reach a specific height at a given angle.
Shadow Length: L = h / tan(α)
L is the length of the shadow, h is the height of the object, and α is the angle of elevation of the sunlight. This helps to find the length of the shadow cast by an object.
Width between two points: W = tan(θ1) × h1 + tan(θ2) × h2
W is the width between two observation points, θ1 and θ2 are the angles of depression to the base of the observations, and h1 and h2 are the heights of the observation points.
Height from Angle of Depression: h = d × tan(θ)
h is the height of an object viewed from an angle of depression θ, and d is the horizontal distance. This is used when assessing heights from above.
Angle of Elevation: tan(θ) = opposite / adjacent
This foundational relationship ties trigonometric functions to right triangles and assists in solving for unknown angles.
Angle of Depression: tan(θ) = opposite / adjacent
Similar to angle of elevation, this applies to angles measured looking downward. Useful for calculating heights or distances from above objects.
Height of Chimney: h = d × tan(θ) + observer height
Calculates the total height of structures by adding observer height at the point of measurement to the result of the tangent function.
Height of Tower: h = d × tan(θ)
Finds the height of a tower using distance from the base (d) and angle of elevation (θ).
Equations
tan(60°) = h / 15
Used in calculating the height of a tower where 15 m is the distance from the base, deriving the equation for h will give h = 15√3.
sin(60°) = 3.7 / L
From the angle measurement on a ladder, this equation is used to calculate the length (L) required when the height is 3.7 m.
tan(45°) = AE / 28.5
This equation helps derive the height of a chimney where AE needs to be solved; results in AE = 28.5.
tan(30°) = 10 / PA
This relates the distance PA to the height of a 10 m building, and solving gives PA = 10√3.
tan(60°) = h / x
This is used in triangles for tower shadow problems where h is the unknown height in terms of shadow length (x).
AB = AE + 1.5
Defines the total height of the chimney in terms of the observer's height and the distance observed.
BC = 40 + x
This describes the relation where distances to shadows switch due to angle variations.
DC / BD = cot(60°)
This equation relates vertical distance to the angle and is key in calculating the foot distance from the ladder to its base.
tan(30°) = PD / AD
This equation helps in solving for the width of a river, using heights and angles observed above.
h = 20√3
This equation denotes the height of a physical object, like a tower, derived from trigonometric ratios and distances.
