Matrix Transformations Lesson 3 Aims: Know the trig exact values for 00, 300, 450, 600 and 900 To use these values to describe transformations. To be able to find a single matrix to describe a combination of transformations.

Trig Exact Values These can be found using some standard triangles. Find the exact value of sin–1 in radians. To solve this, remember the angles whose trigonometric ratios can be written exactly: tan cos sin 90° 60° 45° 30° 0° degrees radians 1 From this table sin–1 =

The Calculator Help Type in the desired value e.g. sin(45) However, your calculator can help you, you just need to good at spotting familiar decimals or using this trick. Otherwise you may have a natural display scientific calculator that does this for you! Type in the desired value e.g. sin(45) Square this answer. Use the fraction button to show this as an exact answer. Then square root the fraction without the calculator. Simplify and you are done! Sin45 0.5 Try cos 210o

Applying to Matrices Reflection in y = x tanθ Exact trig values are useful when dealing with matrix transformations as both reflections and rotations deal with trig values. This will let you describe the matrices more accurately without needing to use decimals or round off numbers. Example 1 Find the matrix of a reflection in the line Reflection in y = x tanθ

On w/b Reflection in y = x tanθ Find the matrix of a reflection in the line Reflection in y = x tanθ

On w/b Reflection in y = x tanθ Find the matrix of a reflection in the line Reflection in y = x tanθ

Rotation θ0 Anticlockwise Applying to Matrices Example 2 Find the matrix of a rotation about the origin 450 anticlockwise. Rotation θ0 Anticlockwise about (0,0) Example 3 Find the matrix of a rotation about the origin 300 clockwise. Try Exercise 4B page 42

Rotation θ0 Anticlockwise On w/b Find the matrix of a rotation about the origin 600 clockwise Rotation θ0 Anticlockwise about (0,0)

Rotation θ0 Anticlockwise On w/b Find the matrix of a rotation about the origin 600 clockwise Rotation θ0 Anticlockwise about (0,0)

Working backwards You may be given the matrix but have to describe the transformation. Remember to use the correct language; reflection, rotation, enlargement, stretch. Remember to give all the details needed. Reflection: Mirror Line Rotation: about the origin, angle, acw/cw Enlargement: Scale Factor Stretch: Direction and Scale Factor

Working backwards Example 4 Find the transformation represented by the matrix

Try match up cards then Exercise 4C page 44 Working backwards Example 5 Find the transformation represented by the matrix Try match up cards then Exercise 4C page 44

Match up!

Match up!

Combinations We can find a s__________ transformation matrix that performs two transformations. This is done by m________________ the two matrix transformations. The o_______ you multiply depends on the order you want to apply the transformations in. This may not matter but quite often does. Transformation Matrix B is applied second. Transformation Matrix A is applied first. B multiplied by A will give you the transformation matrix that performs both transformations in one go.

Combination Transformations Example 1 Find the matrix that represents an enlargement, centre O, scale factor 2 followed by a rotation, centre O, anticlockwise through 900. May find useful Rotation θ0 Anticlockwise about (0,0)

Rotation θ0 Anticlockwise On w/b Find the matrix that represents an enlargement, centre O, scale factor 1/2 followed by a rotation, centre O, anticlockwise through 450. May find useful Rotation θ0 Anticlockwise about (0,0)

Rotation θ0 Anticlockwise On w/b Find the matrix that represents an enlargement, centre O, scale factor 1/2 followed by a rotation, centre O, anticlockwise through 450. May find useful Rotation θ0 Anticlockwise about (0,0)

Combination Transformations Example 2 Find the matrix representing a reflection in the line y = x followed by a stretch scale factor 3 parallel to the x-axis. May find useful Reflection in y=xtanθ

On w/b Reflection in y=xtanθ May find useful Find the matrix representing a reflection in the line y = -x followed by a stretch scale factor 2 parallel to the y-axis. May find useful Try Exercise 4D page 48 qu 1-7 only Be careful with the order you apply them in. Also try to consider if you would get the same result if the order was reversed. Reflection in y=xtanθ

On w/b Reflection in y=xtanθ May find useful Find the matrix representing a reflection in the line y = -x followed by a stretch scale factor 2 parallel to the y-axis. May find useful Try Exercise 4D page 48 Be careful with the order you apply them in. Also try to consider if you would get the same result if the order was reversed. Reflection in y=xtanθ