# In triangle TUV, how do you express CosT in terms of t, u, v?

Jan 12, 2017

$\cos T = \frac{{u}^{2} + {v}^{2} - {t}^{2}}{2 u v}$

#### Explanation:

the standard cosine rule for triangle with vertices $A , B , C$ is:

${a}^{2} = {b}^{2} + {c}^{2} - 2 c b \cos A$

note the angle in the trig. function is the opposite angle to the side on the LHS

so if we had $C o s C$ to find the formula would be:

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos C$

so if we have $\Delta T U V$ and we want $\cos T$ we note that we start with the side $t$ in the formula

${t}^{2} = {u}^{2} + {v}^{2} - 2 u v \cos T$

we now rearrange in terms of $\cos T$

$2 u v \cos T = {u}^{2} + {v}^{2} - {t}^{2}$

$\cos T = \frac{{u}^{2} + {v}^{2} - {t}^{2}}{2 u v}$