Transformations move or resize figures. Trigonometry extends the Pythagorean theorem to find sides and angles in any right triangle.
Translation: slide — (x,y)→(x+a,y+b). Reflection: flip over a line. Rotation: turn around a point. These preserve shape AND size (congruence).
Scale every coordinate from a center: (x,y)→(kx,ky) from origin. Preserves shape but not size — produces similar figures. Scale factor k>1 enlarges, 0<k<1 shrinks.
In a right triangle with angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Memorize: SOH-CAH-TOA.
To find a side: set up ratio and multiply/divide. To find an angle: use inverse trig (sin⁻¹, cos⁻¹, tan⁻¹). Special angles: sin30°=0.5, sin45°≈0.707, sin60°≈0.866.
Q1: In a right triangle, sin θ equals:
Q2: A right triangle has a 30° angle and hypotenuse 20. The opposite side is:
Q3: A dilation with scale factor ½ from the origin maps (6,10) to: