Building blocks

## Suggestion

Can you explain why the points given in the diagram below have coordinates $(1, \tan \theta)$ and $(\cot \theta, 1)$?

You may find it helpful to consider similar triangles, or look at Going round in circles.

Now move the slider for $\delta \theta$.

• What is the relationship between the blue angle and the red angle?

• What can you say about the blue triangles and the red triangle as $\delta \theta$ decreases to $0$?

Take a look at the triangles in the diagram.

What similar triangles can you find?

What do you see as $\delta \theta \rightarrow 0$?

Consider the blue triangle which has one vertex at $(1, \tan \theta)$.

• What is the length of the circular arc drawn through $(1, \tan \theta)$ and subtended by $\delta \theta$?

It may be helpful to label some points in the diagram.

How can the length of the hypotenuse of the red triangle be expressed in terms of $\theta$?

How can this help us to find the arc length?

• Which side length shows the increase in $\tan \theta$ as $\theta$ increases by $\delta \theta$?

$QP$ is $\tan \theta$ and $QT$ is $\tan (\theta+ \delta \theta)$.

How can you use these ideas to find the derivative of $\tan \theta$?

When $\delta \theta$ is small, one side of each blue triangle can be approximated by a circular arc.

As $\delta \theta \rightarrow 0$, the red and blue triangles get closer and closer to being similar. Which are the corresponding sides in the red and blue triangles?