Geometric Bites

The angle–difference identities are: sin(α − β) = sinα cosβ − cosα sinβ cos(α − β) = cosα cosβ + sinα sinβ They can be seen geometrically using a rectangle OZYX with a few right-angled triangles inside it. All side lengths can be written in terms of sinα, cosα, sinβ and cosβ. First, draw a right triangle OPQ with hypotenuse OQ = 1 and angle β at O. By definition: OP = cosβ (horizontal side), QP = sinβ (vertical side). Next, use OP as the hypotenuse of another right triangle OPZ. The right angle is at Z, and the angle at P is α. The hypotenuse is OP = cosβ, so: OZ = sinα cosβ, ZP = cosα cosβ. In a similar way, use QP as the hypotenuse of a right triangle PQY. The right angle is at Y, and the angle at Q is α. The hypotenuse is QP = sinβ, so: QY = cosα sinβ, PY = sinα sinβ. Drop a vertical line from Y to the base at Z, and a horizontal line from Y to the left side at X. This makes OZYX a rectangle with: base OZ, height ZY. On the...

The angle–addition identities for sine and cosine often appear as algebraic formulas, but they can also be understood by combining two right triangles in a simple geometric construction. The calculations for the side lengths follow directly from the definitions of sine and cosine. sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y − sin x sin y Start with a right triangle of angle y and hypotenuse 1. From basic trigonometry, its horizontal and vertical sides are: cos y and sin y. Next, attach a second right triangle with angle x . Its hypotenuse is the side of length cos y from the first triangle, so its adjacent and opposite sides become: adjacent = cos x · cos y opposite = sin x · cos y Likewise, if the first triangle's vertical side sin y is used as a hypotenuse in a similar way, it contributes: adjacent = cos x · sin y opposite = sin x · sin y When the horizontal components are combined, they give the expression for cos(x + y...

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 .