In your terms, Tangent is y/x. Cotangent is x/y.
Take a step up a slope at a particular angle. Sine is how far you moved up, “rise”. Cosine is how far you moved horizontally, “run”. Tangent is rise divided by run; this is a common way of describing a slope — how much your altitude changes if you move a given distance horizontally. Cotangent is run divided by rise (just 1 over the tangent)
The Tangent of an angle gives us the gradient of the line as y/x, this is why tangent can be calculated by sine°/cosine°, the gradient being the rate at which the line the line goes up vs going sidewards. Therefore the cotangent is the reverse, the x/y, the ratio of sidewards vs upwards movement of a line
It blew me away when our maths teacher taught us the unit circle, coz prior to this, our maths teacher from the previous year had taught us trigonometry but never explained to us why we followed soh cah toa, and never explained to us what it meant to find the sin/cos/tan of an angle. If you’re still interested in more trigonometry, look for unit circle videos on YouTube
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