Geometric Bites

3D Rotation Matrix Primer When you rotate a point (x, y, z) in 3D space, you are applying a transformation that changes its coordinates while keeping its distance from the origin the same. This transformation is done using a rotation matrix . 1. Rotation about the x-axis (angle α) Rotation around the x-axis keeps x fixed and rotates the (y, z) plane. (x', y', z') = ( x, y·cos(α) − z·sin(α), y·sin(α) + z·cos(α) ) 2. Rotation about the y-axis (angle β) Rotation around the y-axis keeps y fixed and rotates the (x, z) plane. (x', y', z') = ( x·cos(β) + z·sin(β), y, −x·sin(β) + z·cos(β) ) 3. Rotation about the z-axis (angle γ) Rotation around the z-axis keeps z fixed and rotates the (x, y) plane. (x', y', z') = ( x·cos(γ) − y·sin(γ), x·sin(γ) + y·cos(γ), z ) 4. Combining rotations To rotate a point around all three axes, we combine the rotations. The standard order used in our Desmos cube system is: R = Rz(γ) → R...

In the workings below I demonstrate how you can derive 1+tan²(θ)=sec²(θ) and 1+cot²(θ)=cosec²(θ) using sin²(θ)+cos²(θ)=1 .   Just follow the instructions, and presto!

Below I demonstrate how to derive tan(π/8) , sin(π/8) and cos(π/8) from scratch. These workings are a bit similar to the workings for tan((3π)/8) , sin((3π)/8) and cos((3π)/8) which can be found here : https://geometricbites.blogspot.com/2021/08/how-to-derive-tan38-sin38-and-cos38.html The edge with length R √(2- √(2)) was actually derived in the tan((3π)/8), sin((3π)/8) and cos((3π)/8) post, so doesn't need to be found again. Part 1  Part 2  Part 3  Part 4

Below I demonstrate how to derive tan((3π)/8) , sin((3π)/8) and cos((3π)/8) from scratch. I use geometry and also algebra . Part 1  Part 2 Part 3 Part 4 Part 5  Part 6  If you'd like quick updates every time I post, follow me on Twitter at: https://twitter.com/tiago_hands .