Exercises: sin, cos and tan

Part 1

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

Required: $r$, $s$

First calculate $x$ using the tangent.

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

Solve for $x$ and plug in values.

$r=\frac{25m}{\tan \left({25}^{\circ }\right)}\approx 54m$

Compute $s$ using the sine.

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

Solve for $s$ and plug in values.

$s=\frac{25m}{\sin \left({25}^{\circ }\right)}\approx 59m$

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

Part 2

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

Required: $r$, $s$

First calculate $r$ using the tangent.

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

Solve for $r$ and plug in values.

$r=\frac{25m}{\tan \left({38}^{\circ }\right)}\approx 32m$

Compute $s$ using the sine.

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

Solve for $s$ and plug in values.

$s=\frac{25m}{\sin \left({38}^{\circ }\right)}\approx 41m$

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

Part 3

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

Required: $r$, $s$

First calculate $r$ using the tangent.

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

Solve for $r$ and plug in values.

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

Compute $s$ using the sine.

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

Solve for $s$ and plug in values.

$s=\frac{25m}{\sin \left({45}^{\circ }\right)}\approx 35m$

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