This chapter explores how trigonometry is applied in real-life situations, particularly in measuring heights and distances.
Some Applications of Trigonometry – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Some Applications of Trigonometry chapter of Mathematics. 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
tan θ = opposite / adjacent
θ is the angle of elevation or depression. This ratio helps in finding the height or distance when one of them is known. Used in problems involving right-angled triangles.
sin θ = opposite / hypotenuse
θ is the angle of elevation or depression. This ratio is useful when the hypotenuse and one side are known to find the other side. Applicable in scenarios like ladder problems.
cos θ = adjacent / hypotenuse
θ is the angle of elevation or depression. It's used when the adjacent side and hypotenuse are known to find the opposite side. Common in navigation and surveying.
Angle of Elevation
The angle between the line of sight and the horizontal when the object is above the horizontal level. Essential for determining heights of tall objects like towers.
Angle of Depression
The angle between the line of sight and the horizontal when the object is below the horizontal level. Used in problems involving depths or distances below the observer.
Pythagoras Theorem: a² + b² = c²
a and b are the legs, c is the hypotenuse of a right-angled triangle. It's foundational for solving problems involving distances and heights.
Height of object = Distance × tan θ
θ is the angle of elevation. This formula directly relates the height of an object to the distance from the object and the angle of elevation.
Distance = Height / tan θ
θ is the angle of elevation. Useful for finding the horizontal distance when the height and angle of elevation are known.
Length of shadow = Height / tan θ
θ is the Sun’s altitude. This formula helps in determining the length of the shadow cast by an object under sunlight.
Slope = tan θ
θ is the angle of inclination. It represents the steepness of a line or surface, crucial in construction and engineering.
Equations
Height of tower = Distance from tower × tan(angle of elevation)
Directly calculates the height of a tower or building when the distance from the base and the angle of elevation to the top are known.
Length of ladder = Height to reach / sin(angle of elevation)
Determines the required length of a ladder to reach a certain height at a specific angle, ensuring safety and efficiency.
Distance between two objects = Height difference / tan(angle of depression)
Calculates the horizontal distance between two objects when the height difference and angle of depression from the higher to the lower object are known.
Width of river = Height of bridge × (tan(angle of depression₁) + tan(angle of depression₂))
Used to find the width of a river or any obstacle by measuring angles of depression from a known height.
Height of multi-storeyed building = Height of small building + (Distance between buildings × tan(angle of elevation))
Calculates the height of a taller building by adding the height of a smaller building to the product of the distance between them and the tangent of the angle of elevation.
Shadow length change = Height × (cot(angle₁) - cot(angle₂))
Determines how much the shadow length changes when the Sun’s altitude angle changes, useful in astronomy and timekeeping.
Distance walked towards building = Initial distance - (Height of building / tan(new angle of elevation))
Calculates how much closer a person has walked towards a building by observing the change in the angle of elevation to the top.
Height of pole = Length of rope × sin(angle with ground)
Finds the height of a pole or mast when the length of a rope tied from the top to the ground and the angle it makes with the ground are known.
Length of slide = Height / sin(angle of inclination)
Determines the length of a slide required to achieve a certain height at a specific angle, important in playground design.
Distance from tower = Height of tower / tan(angle of elevation)
Finds how far an observer is from a tower when the height of the tower and the angle of elevation to its top are known.
This chapter explores quadratic equations, highlighting their forms and significance in real-world applications.
This chapter introduces arithmetic progressions, which are sequences of numbers generated by adding a fixed value to the previous term. Understanding these patterns is crucial for solving real-life mathematical problems.
This chapter focuses on the properties of triangles, specifically their similarity and how it can be applied in various real-world contexts.
This chapter covers the concepts of coordinate geometry, including finding distances between points and dividing line segments. Understanding these concepts is essential for solving geometry problems using algebra.
This chapter focuses on the foundational concepts of trigonometry, particularly the relationships between the angles and sides of right triangles.
This chapter explores the properties of circles, particularly focusing on tangents and their relationship with radii and secants.
This chapter focuses on sectors and segments of circles, essential concepts in geometry. Understanding these helps in solving real-life problems related to areas and measurements.
This chapter explores how to find the surface areas and volumes of various solids, including combinations of basic shapes like cubes, cones, cylinders, and spheres, essential for real-world applications.
Statistics is the chapter that deals with the collection, analysis, interpretation, presentation, and organization of data.
This chapter explores the basic concepts and definitions of probability, highlighting its significance in predicting outcomes in uncertain situations.
