1. ## trig

given tanΘ = 2, use trigometric identities to find the exact value of cscΘ(90°-Θ)

thank you

2. Originally Posted by tangerine_tomato
given tanΘ = 2, use trigometric identities to find the exact value of cscΘ(90°-Θ)

thank you
You can observe that $\displaystyle \csc(90^o-\vartheta)=\sec(\vartheta)$. But if you didn't know that, how would you go about solving this problem?

Note 3 things:

$\displaystyle \sec(\varphi)=\frac{1}{\cos(\varphi)}$
$\displaystyle \csc(\varphi)=\frac{1}{\sin(\varphi)}$ and $\displaystyle \sin(\varphi-\vartheta)=\sin(\varphi)\cos(\vartheta)-\cos(\varphi)\sin(\vartheta)$

Thus,

$\displaystyle \csc(90^o-\vartheta)=\frac{1}{\sin(90^o-\vartheta)}=\frac{1}{\sin(90^o)\cos(\vartheta)-\cos(90^o)\sin(\vartheta)}$

Simplifying, we get:

$\displaystyle \csc(90^o-\vartheta)=\frac{1}{\cos(\vartheta)}=\sec(\varthet a)$

Since $\displaystyle \tan(\vartheta)=2$, we can conclude that $\displaystyle \cos(\vartheta)=\frac{1}{\sqrt{5}}$

Thus, $\displaystyle \csc(90^o-\vartheta)=\sec(\vartheta)=\color{red}\boxed{\sqrt {5}}$

Does this make sense?

--Chris

3. Originally Posted by Chris L T521
You can observe that $\displaystyle \csc(90^o-\vartheta)=\sec(\vartheta)$. But if you didn't know that, how would you go about solving this problem?

Note 3 things:

$\displaystyle \sec(\varphi)=\frac{1}{\cos(\varphi)}$
$\displaystyle \csc(\varphi)=\frac{1}{\sin(\varphi)}$ and $\displaystyle \sin(\varphi-\vartheta)=\sin(\varphi)\cos(\vartheta)-\cos(\varphi)\sin(\vartheta)$

Thus,

$\displaystyle \csc(90^o-\vartheta)=\frac{1}{\sin(90^o-\vartheta)}=\frac{1}{\sin(90^o)\cos(\vartheta)-\cos(90^o)\sin(\vartheta)}$

Simplifying, we get:

$\displaystyle \csc(90^o-\vartheta)=\frac{1}{\cos(\vartheta)}=\sec(\varthet a)$

Since $\displaystyle \tan(\vartheta)=2$, we can conclude that $\displaystyle \cos(\vartheta)=\frac{1}{\sqrt{5}}$

Thus, $\displaystyle \csc(90^o-\vartheta)=\sec(\vartheta)=\color{red}\boxed{\sqrt {5}}$

Does this make sense?

--Chris
Two "var"s in one post...you..you've made me proud

4. Originally Posted by Chris L T521
You can observe that $\displaystyle \csc(90^o-\vartheta)=\sec(\vartheta)$. But if you didn't know that, how would you go about solving this problem?

Note 3 things:

$\displaystyle \sec(\varphi)=\frac{1}{\cos(\varphi)}$
$\displaystyle \csc(\varphi)=\frac{1}{\sin(\varphi)}$ and $\displaystyle \sin(\varphi-\vartheta)=\sin(\varphi)\cos(\vartheta)-\cos(\varphi)\sin(\vartheta)$

Thus,

$\displaystyle \csc(90^o-\vartheta)=\frac{1}{\sin(90^o-\vartheta)}=\frac{1}{\sin(90^o)\cos(\vartheta)-\cos(90^o)\sin(\vartheta)}$

Simplifying, we get:

$\displaystyle \csc(90^o-\vartheta)=\frac{1}{\cos(\vartheta)}=\sec(\varthet a)$

Since $\displaystyle \tan(\vartheta)=2$, we can conclude that $\displaystyle \cos(\vartheta)=\frac{1}{\sqrt{5}}$

Thus, $\displaystyle \csc(90^o-\vartheta)=\sec(\vartheta)=\color{red}\boxed{\sqrt {5}}$

Does this make sense?

--Chris
i dont understand that stuff in the 3rd line after the word thus

is that cause of the complementary angle theorem

and i dont understand anything after that

but im too tired ATM to make much effort to understand it

5. Originally Posted by tangerine_tomato
i dont understand that stuff in the 3rd line after the word thus

is that cause of the complementary angle theorem

and i dont understand anything after that

but im too tired ATM to make much effort to understand it
It all boils down to finding $\displaystyle \cos \theta$ from $\displaystyle \tan \theta = 2$.

Half the solution is given above. The other half is $\displaystyle {\color{red}-}\sqrt{5}$.

6. Originally Posted by mr fantastic
It all boils down to finding $\displaystyle \cos \theta$ from $\displaystyle \tan \theta = 2$.

Half the solution is given above. The other half is $\displaystyle {\color{red}-}\sqrt{5}$.
Shoot...darn negative sign...

...wait...how would we know for sure it was in the first quadrant instead of the third quadrant or vice versa? I didn't see any restriction on the angle...

EDIT: Just realized what he meant...never mind...