Half-angle formulae from a known value

Statement

Given that $\cos\theta=\dfrac{7}{25}$ with $0<\theta<90^\circ$, find the exact values of $\sin\dfrac{\theta}{2}$, $\cos\dfrac{\theta}{2}$ and $\tan\dfrac{\theta}{2}$.

Solution

Since $0<\theta<90^\circ$, $\theta/2$ is also in the first quadrant and the functions are positive.

$\sin\dfrac{\theta}{2}=\sqrt{\dfrac{1-\cos\theta}{2}}=\sqrt{\dfrac{1-\frac{7}{25}}{2}}=\sqrt{\dfrac{9}{25}}=\dfrac{3}{5}=0{.}6$.

$\cos\dfrac{\theta}{2}=\sqrt{\dfrac{1+\cos\theta}{2}}=\sqrt{\dfrac{1+\frac{7}{25}}{2}}=\sqrt{\dfrac{16}{25}}=\dfrac{4}{5}=0{.}8$.

$\tan\dfrac{\theta}{2}=\dfrac{3/5}{4/5}=\dfrac{3}{4}=0{.}75$.

$\boxed{\sin\tfrac{\theta}{2}=\tfrac35;\ \cos\tfrac{\theta}{2}=\tfrac45;\ \tan\tfrac{\theta}{2}=\tfrac34}$