Exercises: sin, cos and tan

1
When the evening sun is low, Luise, who is $1.55 m$ tall, casts a shadow $12 m$ 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 $△ A K F$ 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 $△ A K F$ , and
which side is the adjacent cathetus to $α$ ,
and which one is the opposite cathetus to $α$ .
Determining hypotenuse, adjacent and opposite cathetus
Hypotenuse: side [AK] with $AK = ?_{}$
adjacent cathetus to $α$ : side [FA] with $AK = ?_{}$
opposite cathetus to $α$ : side [FK] with $AK = ?_{}$
The formulas for sin, cos and tan are:
$\sin α = \frac{oppositecathetus}{hypotenuse}$
$\cos α = \frac{adjacent cathetus}{hypotenuse}$
$\tan α = \frac{opposite cathetus}{adjacent cathetus}$
Since the hypotenuse is not given, it should preferably not appear in the formula you use.
$\to$ Solve the problem with the tangent.
Solution
$\tan α = \frac{opposite cathetus to α}{adjacent cathetus to α}$
Plug in the lengths from the task into this formula.
$\tan α = \frac{FK}{FA}$
Now insert the numerical values for the lengths of the distances and calculate the tangent of the angle.
$\tan α = \frac{1.55 m}{12 m} = \frac{31}{240} \approx 0.129$
To obtain the angle $α$ , apply the inverse function $\tan^{- 1}$ with the calculator.
$α = \tan^{- 1} (\frac{31}{240}) \approx {7.4}^{\circ}$
⇒ The sun's rays fall on the road at an angle of ${7.4}^{\circ}$ .
2
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.

For this task you need the following basic knowledge: Sine, Cosine and Tangent
$\tan (31^{\circ})$ $=$ $\frac{h}{20 m}$ $\cdot 20 m$
$h$ $=$ $\tan (31^{\circ}) \cdot 20 m$
$h$ $=$ $12 m$
$\Rightarrow$ The height of the fir tree is $12 m$ .
3
The drawbridge of a castle is $8 m$ 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?

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Determine the chain length $k$ using the sine.
$\sin (43 °)$ $=$ $\frac{8 m}{k}$ $\cdot k$
$\sin (43 °) \cdot k$ $=$ $8 m$ $: \sin (43 °)$
$k$ $=$ $\frac{8 m}{\sin (43 °)}$
$k$ $\approx$ $11.7 m$
The chain must be about $11.7 m$ long so that you can lower the drawbridge.
4
To determine the width of a river, one has marked out the distance $AB = 80 m$ on one bank. On the other bank, opposite $B$ , there is a point $C$ . The angle between $A B$ and $A C$ is measured as $α = 38^{\circ}$ . First make a sketch and then calculate the width of the river.

For this task you need the following basic knowledge: Sine, Cosine and Tangent
1. Make a sketch
2. Compute the width
$\tan (38^{\circ})$ $=$ $a : 80 m$ $\cdot 80 m$
$a$ $=$ $\tan (38^{\circ}) \cdot 80 m$
$a$ $=$ $62.5 m$
Answer: The river is $62,5 m$ wide
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 $= 100 m$ , $α = 50^{\circ}$
First, draw a sketch.
Use the sine to compute $h$ .
$\sin (50^{\circ}) = \frac{h}{100 m} | \cdot 100 m$
Solve for $h$ .
$h = \sin (50^{\circ}) \cdot 100 m^{}$
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 $100 m$ long line, the flight altitude of $20 m$ 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 $= 100 m$ , opposite cathetus $= 20 m$
First, draw a sketch.
Use the sine of $α$ .
$\sin (α) = \frac{20 m}{100 m}$
Compute $α$ .
$α = {11.5}^{\circ}$
$\Rightarrow$ The angle of ascent must not exceed $11.5 °$ .
7
To bridge a storey height of $3.20 m$ with a staircase, $4.50 m$ are available for the projection.
At what angle of inclination is the stair stringer to be cut?

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Sine, Cosine and Tangent
Given.: $h = 3.2 m; x = 4.5 m$
Required.: $α$
Make a sketch.
Calculate the angle $α$ using the tangent.
$\tan (α)$ $=$ $\frac{h}{x}$
↓
plug in
$\tan (α)$ $=$ $\tan (α) = \frac{3.2}{4.5}$
↓
compute $α$ using $\tan^{- 1}$ .
$α$ $=$ $\tan^{- 1} (\frac{3.2}{4.5})$
$α$ $=$ ${35.4}^{\circ}$
$\Rightarrow$ The stair scale shall be cut at an inclination angle of $α = 35.4 °$ .
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 $h = 25 m$ .
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: $h = 25 m$ ; $α = 25^{\circ}$
Required: $r$ , $s$
First calculate $x$ using the tangent.
$\tan (α) = \frac{h}{r}$
Solve for $x$ and plug in values.
$r = \frac{25 m}{\tan (25^{\circ})} \approx 54 m$
Compute $s$ using the sine.
$\sin (α) = \frac{h}{s}$
Solve for $s$ and plug in values.
$s = \frac{25 m}{\sin (25^{\circ})} \approx 59 m$
$\Rightarrow$ The projection is $54 m$ and the stair length is $59 m$ .
Part 2
Given: $h = 25 m$ ; $α = 38^{\circ}$
Required: $r$ , $s$
First calculate $r$ using the tangent.
$\tan (α) = \frac{h}{r}$
Solve for $r$ and plug in values.
$r = \frac{25 m}{\tan (38^{\circ})} \approx 32 m$
Compute $s$ using the sine.
$\sin (α) = \frac{h}{s}$
Solve for $s$ and plug in values.
$s = \frac{25 m}{\sin (38^{\circ})} \approx 41 m$
$\Rightarrow$ The projection is $32 m$ and the stair length is $41 m$ .
Part 3
Given: $h = 25 m$ ; $α = 45^{\circ}$
Required: $r$ , $s$
First calculate $r$ using the tangent.
$\tan (α) = \frac{h}{r}$
Solve for $r$ and plug in values.
$r = \frac{25 m}{\tan (45^{\circ})} = 25 m$
Compute $s$ using the sine.
$\sin (α) = \frac{h}{s}$
Solve for $s$ and plug in values.
$s = \frac{25 m}{\sin (45^{\circ})} \approx 35 m$
$\Rightarrow$ The projection is $25 m$ and the stair length is $35 m$ .
9
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.

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Given: a = 18 m; b = 8 m; $α$ = 50°
Required: h
First draw a sketch.
Calculate $x$ by subtracting $b$ from $a$ and halving the result.
$x = \frac{18 m - 8 m}{2} = 5 m$
Calculate $h$ using the tangent.
$\tan (α)$ $=$ $\frac{h}{x}$
↓
Solve for h and plug in the values.
$h$ $=$ $5 m \cdot \tan (50^{\circ})$
↓
compute
$h$ $=$ $6 m$
$\Rightarrow$ The dam height is 6 m.
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.
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: $h = 25 m$ ; $α = 25^{\circ}$
Required: $x$ , $w$
First calculate $x$ using the tangent.
$\tan (α) = \frac{h}{x}$
Solve for $x$ and plug in values.
$x = \frac{25 m}{\tan (25^{\circ})} \approx 54 m$
Compute $w$ using the sine.
$\sin (α) = \frac{h}{w}$
Solve for $w$ and plug in values.
$w = \frac{25 m}{\sin (25^{\circ})} \approx 59 m$
$\Rightarrow$ The projection is 54m and the stair length is 59m.

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: $h = 25 m$ ; $α = 38^{\circ}$
Required: $x$ , $w$
First calculate $x$ using the tangent.
$\tan (α) = \frac{h}{x}$
Solve for $x$ and plug in values.
$x = \frac{25 m}{\tan (38^{\circ})} \approx 32 m$
Compute $w$ using the sine.
$\sin (α) = \frac{h}{w}$
Solve for $w$ and plug in values.
$w = \frac{25 m}{\sin (38^{\circ})} \approx 41 m$
$\Rightarrow$ The projection is $32 m$ and the stair length is $41 m$ .

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: $h = 25 m$ ; $α = 45^{\circ}$
Required: $x$ , $w$
First calculate $x$ using the tangent.
$\tan (α) = \frac{h}{x}$
Solve for $x$ and plug in values.
$x = \frac{25 m}{\tan (45^{\circ})} = 25 m$
Compute $w$ using the sine.
$\sin (α) = \frac{h}{w}$
Solve for $w$ and plug in values.
$w = \frac{25 m}{\sin (45^{\circ})} \approx 35 m$
⇒ The projection is $25 m$ and the stair length is $35 m$ .

$\tan (31^{\circ})$	$=$	$\frac{h}{20 m}$	$\cdot 20 m$
$h$	$=$	$\tan (31^{\circ}) \cdot 20 m$
$h$	$=$	$12 m$

$\sin (43 °)$	$=$	$\frac{8 m}{k}$	$\cdot k$
$\sin (43 °) \cdot k$	$=$	$8 m$	$: \sin (43 °)$
$k$	$=$	$\frac{8 m}{\sin (43 °)}$
$k$	$\approx$	$11.7 m$

$\tan (38^{\circ})$	$=$	$a : 80 m$	$\cdot 80 m$
$a$	$=$	$\tan (38^{\circ}) \cdot 80 m$
$a$	$=$	$62.5 m$

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

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

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