Given: $h=25m$; $\alpha =25^\circ$

Required: $x$, $w$

First calculate $x$ using the tangent.

$\tan\left(\alpha\right)=\frac{h}{x}$

Solve for $x$ and plug in values.

$x=\frac{25m}{\tan\left(25^{\circ}\right)}\approx54m$

Compute $w$ using the sine.

$\sin\left(\alpha\right)=\frac{h}{w}$

Solve for $w$ and plug in values.

$w=\frac{25m}{\sin\left(25^{\circ}\right)}\approx59m$

$\Rightarrow\;\;$The projection is 54m and the stair length is 59m.

Given: $h=25m$; $\alpha =38^\circ$

Required: $x$, $w$

First calculate $x$ using the tangent.

$\tan\left(\alpha\right)=\frac{h}{x}$

Solve for $x$ and plug in values.

$x=\frac{25m}{\tan\left(38^{\circ}\right)}\approx32m$

Compute $w$ using the sine.

$\sin\left(\alpha\right)=\frac{h}{w}$

Solve for $w$ and plug in values.

$w=\frac{25m}{\sin\left(38^{\circ}\right)}\approx41m$

$\Rightarrow\;\;$The projection is $32m$ and the stair length is $41m$.

Given: $h=25m$; $\alpha =45^\circ$

Required: $x$, $w$

First calculate $x$ using the tangent.

$\tan\left(\alpha\right)=\frac{h}{x}$

Solve for $x$ and plug in values.

$x=\frac{25m}{\tan\left(45^{\circ}\right)}=25m$

Compute $w$ using the sine.

$\sin\left(\alpha\right)=\frac{h}{w}$

Solve for $w$ and plug in values.

$w=\frac{25m}{\sin\left(45^{\circ}\right)}\approx35m$

⇒ The projection is $25m$ and the stair length is $35m$.