Copy the diagram and show how sec θ, csc θ, and cot θ relate to the unit circle.
The representation of the diagram is shown if Figure 1. There's a relationship between sec θ, csc θ, and cot θ related the unit circle. Lines green, blue and pink show the relationship.
a.1 First, find in the diagram a segment whose length is sec θ.
The segment whose length is sec θ is shown in Figure 2, this length is the segment [tex]\overline{OF}[/tex], that is, the line in green.
a.2 Explain why its length is sec θ.
We know these relationships:
(1) [tex]sin \theta=\frac{\overline{BD}}{\overline{OB}}=\frac{\overline{BD}}{r}=\frac{\overline{BD}}{1}=\overline{BD}[/tex]
(2) [tex]cos \theta=\frac{\overline{OD}}{\overline{OB}}=\frac{\overline{OD}}{r}=\frac{\overline{OD}}{1}=\overline{OD}[/tex]
(3) [tex]tan \theta=\frac{\overline{FD}}{\overline{OC}}=\frac{\overline{FC}}{r}=\frac{\overline{FC}}{1}=\overline{FC}[/tex]
Triangles ΔOFC and ΔOBD are similar, so it is true that:
[tex]\frac{\overline{FC}}{\overline{OF}}= \frac{\overline{BD}}{\overline{OB}} [/tex]
∴ [tex]\overline{OF}= \frac{\overline{FC}}{\overline{BD}}= \frac{tan \theta}{sin \theta}= \frac{1}{cos \theta} \rightarrow \boxed{sec \theta= \frac{1}{cos \theta}}[/tex]
b.1 Next, find cot θ
The segment whose length is cot θ is shown in Figure 3, this length is the segment [tex]\overline{AR}[/tex], that is, the line in pink.
b.2 Use the representation of tangent as a clue for what to show for cotangent.
It's true that:
[tex]\frac{\overline{OS}}{\overline{OC}}= \frac{\overline{SR}}{\overline{FC}} [/tex]
But:
[tex]\overline{SR}=\overline{OA}[/tex]
[tex]\overline{OS}=\overline{AR}[/tex]
Then:
[tex]\overline{AR}= \frac{1}{\overline{FC}}= \frac{1}{tan\theta} \rightarrow \boxed{cot \theta= \frac{1}{tan \theta}}[/tex]
b.3 Justify your claim for cot θ.
As shown in Figure 3, θ is an internal angle and ∠A = 90°, therefore ΔOAR is a right angle, so it is true that:
[tex]cot \theta= \frac{\overline{AR}}{\overline{OA}}=\frac{\overline{AR}}{r}=\frac{\overline{AR}}{1} \rightarrow \boxed{cot \theta=\overline{AR}}[/tex]
c. find csc θ in your diagram.
The segment whose length is csc θ is shown in Figure 4, this length is the segment [tex]\overline{OR}[/tex], that is, the line in green.
