**Problem 4:**Find the image of the point \((2, 5)\) after a rotation of 90 degrees about the origin. Then rotate the image 30 degrees. Show that the result is the same as rotating the original point, \((2, 5)\), 120 degrees about the origin.

Given coordinate point :- 2,5Rotation formula for angle θx,y→xcosθ-ysinθ,xsinθ+ycosθ When (2,5) rotates 90 degrees about origin Thenθ=90° ⇒2,5→2cos90°-5sin90°,2sin90°+5cos90° Here cos90°=0,sin90°=1 ⇒2,5→20-51,21+50 ⇒2,5→-5,2 Now image (-5,2) rotates about origin by 30 degrees ⇒-5,2→-5cos30°-2sin30°,-5sin30°+2cos30° Here cos30°=32,sinθ=12 ⇒-5,2→-532-212,-512+232 ⇒-5,2→-532-1,-52+3..............................................(i)Now when (2,5) rotates about origin 120 degrees ⇒2,5→2cos120°-5sin120°,2sin120°+5cos120° ⇒2,5→2cos180°-60°-5sin180°-60°,2sin180°-60°+5cos180°-60° Here cos180°-θ=-cosθ,sin180°-θ=sinθ ⇒2,5→-2cos60°-5sin60°,2sin60°-5cos60° Here cos60°=12,sin60°=32 ⇒2,5→-212-532,232+512 ⇒2,5→-1-532,3-52......................................(ii) Result (i) and (ii) both are same Hence The image of point (2,5) after rotation of 90 degrees about origin and then rotates the image 30 ° is the same as rotating the original point (2,5) ,120° about the origin