Exercises: sin, cos and tan

1
When the evening sun is low, Luise, who is $1.55 \text{ m}$ tall, casts a shadow $12 \text{ m}$ long on a level road. Draw a sketch and calculate the angle at which the sunbeam hits the ground.
For this task you need the following basic knowledge: Sine, Cosine and Tangent
Sketch
First thoughts
The triangle $\triangle AKF$ has a right angle at $F$ .
Therefore, the formulas for sin, cos and tan apply to it.
To know which of the formulas you should use, first determine,
which side is the hypotenuse in the triangle $\triangle AKF$ , and
which side is the adjacent cathetus to $\alpha$ ,
and which one is the opposite cathetus to $\alpha$ .
Determining hypotenuse, adjacent and opposite cathetus
Hypotenuse: side [AK] with $\overline{\mathrm{AK}}=?_{ }$
adjacent cathetus to $\alpha$ : side [FA] with $\overline{\mathrm{AK}}=?_{ }$
opposite cathetus to $\alpha$ : side [FK] with $\overline{\mathrm{AK}}=?_{ }$
The formulas for sin, cos and tan are:
$\sin\alpha=\frac{\text{oppositecathetus}}{\text{hypotenuse}}$
$\cos\alpha=\frac{\text{adjacent cathetus}}{\text{hypotenuse}}$
$\tan\alpha=\frac{\text{opposite cathetus}}{\text{adjacent cathetus}}$
Since the hypotenuse is not given, it should preferably not appear in the formula you use.
$\rightarrow$ Solve the problem with the tangent.
Solution
$\tan\alpha=\frac{\text{opposite cathetus to }\alpha}{\text{adjacent cathetus to}\alpha}$
Plug in the lengths from the task into this formula.
$\tan\alpha=\frac{\overline{\mathrm{FK}}}{\overline{\mathrm{FA}}}$
Now insert the numerical values for the lengths of the distances and calculate the tangent of the angle.
$\tan \alpha = \frac {1.55\, \mathrm {m}}{12 \,\mathrm{m}}=\frac{31}{240}\approx 0.129$
To obtain the angle $\alpha$ , apply the inverse function $\tan^{-1}$ with the calculator.
$\alpha=\tan^{-1}(\frac{31}{240})\approx7.4^{\circ}$
⇒ The sun's rays fall on the road at an angle of $7.4^\circ$ .

A fir tree casts a shadow $20 m$ long. The sun's rays hit the earth at an angle of $31^\circ$ . Draw a sketch and calculate the height of the fir tree.

The drawbridge of a castle is $8m$ long and has an angle of $43^\circ$ between the wall and the chain. How long must a chain be that can be used to fold down the drawbridge?

To determine the width of a river, one has marked out the distance $\overline{\mathrm{AB}}=80m$ on one bank. On the other bank, opposite $B$ , there is a point $C$ . The angle between $AB$ and $AC$ is measured as $\alpha=38^\circ$ . First make a sketch and then calculate the width of the river.

5
A hang glider is being towed by a motorboat. From the boat, Till estimates the angle of ascent of the $100 m$ long, tightly stretched tow line to be about $50°$ .
How high is the glider above the water?

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Given: hypothenuse $=100\;m$ , $\alpha=50^{\circ}$
First, draw a sketch.
Use the sine to compute $h$ .
$\sin\left(50^\circ\right)=\dfrac h{100\ \text{m}} \quad \quad \left|\cdot100\ \text{m}\right.$
Solve for $h$ .
$h=\sin\left(50^{\circ}\right)\cdot100\ \text{m}^{ }$
Multiply.
$h=76.6\;m$
$\Rightarrow\;\;$ The glider is $76{,}6\; m$ above the water.
6
For a glider, "flying" behind the motorboat on a $100m$ long line, the flight altitude of $20m$ should not be exceeded for safety reasons.
How large may the angle of ascent of the line maximally be?

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Given: hypothenuse $= 100\ \text{m}$ , opposite cathetus $=20m$
First, draw a sketch.
Use the sine of $\alpha$ .
$\sin\left(\alpha\right)=\frac{20\ \text{m}}{100\ \text{m}}$
Compute $\alpha$ .
$\alpha=11.5^{\circ}$
$\Rightarrow\;\;$ The angle of ascent must not exceed $11.5°$ .

To bridge a storey height of $3.20m$ with a staircase, $4.50m$ are available for the projection.

At what angle of inclination is the stair stringer to be cut?

8
According to industrial standards, the inclination angle of stairs should be $25°-38°$ for main stairs and $38°-45°$ for secondary stairs.
The storey height is $h=25m$ .
How long are the run ( $r$ ) and the stair stringer ( $s$ ) for
25°
38°
45°
?
For this task you need the following basic knowledge: Sine, Cosine and Tangent
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.
$r=\frac{25m}{\tan\left(25^{\circ}\right)}\approx54m$
Compute $s$ using the sine.
$\sin\left(\alpha\right)=\frac{h}{s}$
Solve for $s$ and plug in values.
$s=\frac{25m}{\sin\left(25^{\circ}\right)}\approx59m$
$\Rightarrow\;\;$ 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.
$r=\frac{25m}{\tan\left(38^{\circ}\right)}\approx32m$
Compute $s$ using the sine.
$\sin\left(\alpha\right)=\frac{h}{s}$
Solve for $s$ and plug in values.
$s=\frac{25m}{\sin\left(38^{\circ}\right)}\approx41m$
$\Rightarrow\;\;$ 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.
$s=\frac{25m}{\sin\left(45^{\circ}\right)}\approx35m$
$\Rightarrow\;\;$ The projection is $25m$ and the stair length is $35m$ .

A dam with a trapezoidal cross-section is to be built $50 m$ long. It is to be $18 m$ wide at the bottom and $8 m$ wide at the top. The angle of slope should be $50°$ .

Calculate the height of the dam.

10
According to industrial standards, the inclination angle of stairs should be $25°-38°$ for main stairs and $38°-45°$ for secondary stairs.
The storey height is 25m.
1. How long is the stair stringer for $25°$ . Also calculate the projection.
  
  For this task you need the following basic knowledge: Sine, Cosine and Tangent
  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.
  Do you have a question?
  Write it here 👉
2. How long is the stair stringer for 38°. Also calculate the projection.
  
  For this task you need the following basic knowledge: Sine, Cosine and Tangent
  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 $41m$ .
  Do you have a question?
  Write it here 👉
3. How long is the stair stringer for 45°. Also calculate the projection.
  
  For this task you need the following basic knowledge: Sine, Cosine and Tangent
  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 $35m$ .
  Do you have a question?
  Write it here 👉

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$\displaystyle \tan\left(31^{\circ}\right)$	$=$	$\displaystyle \frac{h}{20m}$	$\displaystyle \cdot20m$
$\displaystyle h$	$=$	$\displaystyle \tan\left(31^{\circ}\right)\cdot20m$
$\displaystyle h$	$=$	$\displaystyle 12m$

$\displaystyle \sin\left(43°\right)$	$=$	$\displaystyle \frac{8\text{m}}{k}$	$\displaystyle \cdot k$
$\displaystyle \sin\left(43°\right)\cdot k$	$=$	$\displaystyle 8\ \text{m}\$	$\displaystyle \ :\sin\left(43°\right)$
$\displaystyle k$	$=$	$\displaystyle \frac{8\text{m}}{\sin\left(43°\right)}$
$\displaystyle k$	$≈$	$\displaystyle 11.7\text{m}\$

$\displaystyle \tan\left(38^{\circ}\right)$	$=$	$\displaystyle a:80\,m$	$\displaystyle ⋅80m$
$\displaystyle a$	$=$	$\displaystyle \tan\left(38^\circ\right)\cdot80\,m$
$\displaystyle a$	$=$	$\displaystyle 62.5\,m$

$\displaystyle \tan\left(\alpha\right)$	$=$	$\displaystyle \frac{h}{x}$
	↓	plug in
$\displaystyle \tan\left(\alpha\right)$	$=$	$\displaystyle \tan\left(\alpha\right)=\frac{3.2}{4.5}$
	↓	compute $\alpha$ using $\tan^{-1}$ .
$\displaystyle \alpha$	$=$	$\displaystyle \tan^{-1}\left(\frac{3.2}{4.5}\right)$
$\displaystyle \alpha$	$=$	$\displaystyle 35.4^{\circ}$

$\displaystyle \tan\left(\alpha\right)$	$=$	$\displaystyle \frac{h}{x}$
	↓	Solve for h and plug in the values.
$\displaystyle h$	$=$	$\displaystyle 5\,\mathrm{m}\cdot\tan\left(50^\circ\right)$
	↓	compute
$\displaystyle h$	$=$	$\displaystyle 6\,\mathrm{m}$

Exercises: sin, cos and tan

Sketch

First thoughts

Determining hypotenuse, adjacent and opposite cathetus

Solution

1. Make a sketch

2. Compute the width

Sine, Cosine and Tangent

Part 1

Part 2

Part 3

Do you have a question?

Do you have a question?

Do you have a question?