## Pre-Requisires

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**Some Applications Of Trigonometry** | **Speed Notes**

**Notes For Quick Recap**

**Introduction :**

One of the main applications of trigonometry is to find the distance between two or more than two places or to find the height of the object or the angle subtended by any object at a given point without actually measuring the distance or heights or angles.

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Trigonometry is useful to astronomers, navigators, architects and surveyors etc. in solving problems related to heights and distances.

**The directions of the objects can be described by measuring :**

(i) angle of elevation and (ii) angle of depression

Angles of elevation or angles of depression of the objects are measured by an instrument called The odolite.

The odolite is based on the principles of trigonometry, which is used for measuring angles with a rotating telescope.

In 1856, Sir George Everest first used the giant theodolite, which is now on display in the Museum of the survey of India in Dehradun.

**Angle of Elevation:**

Let P be the position of the object above the horizontal line OA and O be the eye of the observer, then angle AOP is called angle of elevation. It is called the angle of elevation, because the observer has to elevate (raise) his line of sight from the horizontal OA to see the object P. [ When the eye turns upwards above the horizontal line.]

**Angle of Depression:**

Let P be the position of the object below the horizontal line OA and O be the eye of the observer, then angle AOP is called angle of depression.

It is called the angle of depression because the observer has to depress (lower) his line of sight from the horizontal OA to see the object P.

[When the eye turns downwards below the horizontal line].

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- Real Numbers | Study
- Polynomials | Study
- Pair of Linear Equations in Two Variables | Study
- Quadratic Equations | Study
- Arithmetic Progressions| Study
- Triangles | Study
- Coordinate Geometry | Study
- Introduction To Trigonometry | Study
- Some Applications Of Trigonometry| Study
- Circles | Study
- Areas Related to Circles | Study
- Surface Areas and Volumes | Study
- Statistics | Study
- Probability | Study
- Cuboid And Cube

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## Pre-Requisires

Test & Enrich

**Speed Notes**

**Notes For Quick Recap**

**Cartesian System**

A plane formed by two number lines, one horizontal

and the other vertical, such that they intersect each

other at their zeroes, and then they form a Cartesian

Plane.

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**Coordinate Axes:**

The position of a point in a plane is fixed by selecting the axes of reference which are formed by two number lines intersecting each other at right angles, so that their zeroes coincide.

The horizontal number line is called **x-axis** and vertical number line is called **y axis**.

A point that lies on X Axis is (x,0)

A point that lies on Y Axis is (0,y)

Equation of Y Axis is x = 0

Equation of X Axis is y = 0

Equation of a lne parallel to Y Axis is x = a

Equation of a lne parallel to X Axis is y = a

Equation of a lne perpendicular to X Axis is X = a

Equation of a lne perpendicular to X Axis is X = a

The point of intersection of the two lines is called **origin**.

is the **x-axis** and Y1OY is the **y-axis**. These coordinate axes are also called **rectangular axes** as they are **perpendicular** to each other.

Rectangular coordinates are **ordered pairs** in which the first element is called the **abscissa** and the second element is called the **ordinate**.

● In the **first quadrant**, x is + ve and y is + ve

● In the **second quadrant**, x is – ve and y is + ve

● In the** third quadrant**, x is – ve and y is – ve

● In the **fourth quadrant**, x is + ve and y is -ve.

**Distance Formula:**

**Example:**

**Example:**

**Collinearity of three points:**

Three points P, Q and R are said to be collinear, if they lie in the same straight line.

i.e., PR = PQ + QR

i.e., PQ = PR + RQ

i.e., QR = QP + PR

If three points are not collinear, they always form a triangle.

**Special Polygons:**

**(i) In Case of Triangle**

(a) a right-angled triangle, if sum of squares of any two sides is equal to square of third side.

(b) an equilateral triangle, if all the three sides are equal.

(c) an isosceles triangle, if any two sides are equal.

**(ii) In Case of Quadrilateral**

(a) parallelogram, if opposite sides are equal and diagonals are not equal.

(b) rectangle, if opposite sides are equal and diagonals are equal.

(c) square, if all the four sides are equal and diagonals are equal.

(d) rhombus, if all the four sides are equal and diagonals are not equal.

**Section Formula (Internal division only)**

**Midpoint Formula:**

**Point Dividing Two points in K : 1 Ratio:**

**Note:**

If k is positive, the point divides the given points internally.

If k is Negative, the point divides the given points externally

**Coordinates of the centroid of a triangle:**

**Points of Trisection:**

If a line segment is divided into three equal parts by two points, the points are said to be the **points of trisection**.

In the given figure, the points R and S divide the line segment PQ into three equal parts i.e., PR=RS=SQ. The points R and S are said to be points of trisection.

**Area of a Triangle:**

The area of the triangle formed by the points

is calculated by the following expression.

Area of ∆PQR =

**Area of Quadrilateral:**

Area of a quadrilateral can be found by splitting up the quadrilateral into two triangles and sum up their areas.

Thus, area of quadrilateral PQRS = area of ∆PQR+ area of ∆PRS

**Condition for collinearity of three points :**

Three given points will be collinear, if the area of the triangle formed by these points is zero.

Rule to prove that three given points are collinear:

**Step 1.** Find the area of the triangle formed by the given points.

**Step 2.** Show that the area of the triangle formed by the given points is zero.

* The coordinates of the origin are O(0,0)

* The coordinates of any point on x-axis are (x, 0)

i.e., y=0 or ordinate is zero.

* The coordinates of any point on y – axis are (0, y) i.e., x=0 or abscissa is zero.

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- Cuboid And Cube
- Probability | Study
- Statistics | Study
- Surface Areas and Volumes | Study
- Areas Related to Circles | Study
- Circles | Study
- Some Applications Of Trigonometry| Study
- Introduction To Trigonometry | Study
- Coordinate Geometry | Study
- Triangles | Study
- Arithmetic Progressions| Study
- Quadratic Equations | Study
- Pair of Linear Equations in Two Variables | Study
- Polynomials | Study
- Real Numbers | Study

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