Question

If SecO = 13/12 and O is between 3pi/2 and 2pi, find the exact values in reduced form for:A.)Tan OB.)Sin(20) C.)Cos(O/2)

Answer 1

Identify and Set Up

We are given a question on trigonometric identities.

Our approach is to:

First, we resolve the identity, sec,

- Get the sides and angles of our triangles

- Resolve the other identities.

Execute

`\sec O=(1)/(\cos O)=(h)/(a)=(13)/(12)`

We now have for ourselves a right angled triangle with hypothenuse 13 and adjacent 12.

From this, we know that the opposite side is:

`x=\sqrt[]{13^2-12^2}=\sqrt[]{25}=5`

A)

Tan O

`\tan 0=\frac{\text{opp}}{\text{hyp}}=(5)/(12)`

B)

Sin2O

`\sin 2O=2\sin O\cos O`

Hence, we have to find sin O and cos O

`\begin{gathered} \sin O=\frac{\text{opp}}{\text{hyp}}=(5)/(13) \\ \cos O=\frac{\text{adj}}{\text{hyp}}=(12)/(13) \end{gathered}`

Hence, we have: