]> Exercise 2.6

Exercise 2.6

Derive the relations between these functions by using similar triangles.

Solution:

By similar triangles, the ratio of BD to BO, sin θ to 1 is the same as the ratio of CB to OC, tan θ to sec θ , and of OB to AO (1 to csc θ ), and of CO to AC ( s e c θ to ( tan θ + cot θ )), and of CD to BC (( sec θ cos θ ) to tan θ ), and of OE to BO ( sin θ to 1)

This tells us

csc θ = 1 sin θ

so that on taking complements we have as well,

sec θ = 1 cos θ

We also get

tan θ = sin θ sec θ = sin θ cos θ

and the complementary result

cot θ = cos θ sin θ