Tangent
When we looked at sine, we looked at a triangle in a circle. The word sine is a mis-translation of chord and trigonometry began with triangles in circles. The word tangent means to touch. A line that touches a circle in just one point is tangent to that circle.
The reason we look at trigonometry within circles is because trigonometry was developed for measuring the distances and angles between stars in Astronomy and in navigation. As stars appear to move in the night’s sky, they create an arc, or part of a circle, in the sky as they are viewed from Earth. The circle in our diagrams could represent the movement of the star and triangles are formed by drawing lines between the centre of the circle and the position of stars.
Here we add the radius of the circle at right angles to the tangent
Now we have a right angled triangle.
We can make a smaller, similar triangle within the circle
Let’s say the side of the hypotenuse of the smaller triangle is 1 and label the angle at the centre of the circle as θ.
So, now we have these two similar triangles, the smaller blue one and the one extended by the red lines
Looking at the smaller blue triangle and using the trigonometric ratios we know
sin θ = O ÷ H
In the blue triangle H = 1, so sin θ = O.
cos θ = A ÷ H but in this triangle, as H = 1, cos θ = A.
Replacing sides O and A in the blue triangle above with the sin θ and cos θ, and labelling the tangent as tan θ and labelling the points of the two diagrams, we get the picture here.
Using our knowledge of similar triangles, we know that
DE ÷ AE = BC ÷ AC
AE is 1 because it is the radius of the circle.
DE = BC ÷ AC
DE is tan θ, BC is sin θ, AC is cos θ
So, tan θ = sin θ ÷ cos θ
