**x.y coordinate system of rotation**: Let** x y coordinate system** be given. We

We rotate the x y coordinate system about the origin through an angle θ, where 0 < θ < 90°, resulting in the new axes O. X and O. Y. Consider a point P with coordinates (x, y) in the x y system. Let (X, Y) be the coordinates of point P in the X Y coordinate system. We need to find X and Y in terms of the given coordinates x and y. α represents the angle that OP makes with Ox.

about the **origin** through an **angle θ** means** 0< θ < 90°** so that the **new axis are O.X and O.Y** as shown in the diagram. Let a point** P has coordinated (x, y)** referred to the **x y system of coordinates**. Suppose** p has coordinated (X, Y)** referred to the** X Y coordinate system**. We have to fined** X, Y in terms of the given coordinate x, y.** Let** α be a measure** of the angle that** OP makes Ox.**

x y coordinate system of rotation proof:

**From P**, draw** PM perpendicular to ox and PM **and **perpendicular to Ox.**

**Let | OP | =r**. From the **right angle ∆ OPM**‘ we have

**OM’ =X = r cos (α – θ)………………..(1a)**

**M’P= Y= r cos (α – θ)………………..(1b)**

these equations can be written as

**X = r cos α cos θ + r sin α sin θ**

**Y = r cos α cos θ – r sin α sin θ**

Also, from the **∆ OPM** we have

**x = r cos α y = r sin α ………..(2)**

Equation 2 can be written as

**r = x / cos α and r = y / sin α**

Putting the value of 2 in equation 1 we get the result

**X = x cos θ + y sin θ**

**Y = y cos θ – x sin θ**

that become

**(X, Y) = (x cos θ + y sin θ, y cos θ – x sin θ)**

are the** coordinate of P** referred to the **new axis OX and O Y.**

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