Exercises: Sine, Cosine and Tangent on a right triangle

Calculate the missing sides and angles of the isosceles triangle $A B C$ with $a = b$ . Note that we are looking at general isosceles triangles that are not necessarily right-angled.

a=44.2cm
c=63.4cm

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Given: a=b= 44.2cm c=63.4cm
Required: h, $α, β, γ$
Draw a sketch for an overview.
First compute $x$ .
$x$ $=$ $\frac{c}{2}$
$x$ $=$ $\frac{63.4 c m}{2}$
$=$ $31.7 c m$
Compute $h$ by using the Pythagorean theorem within the triangle $△ D B C$ .
$h$ $=$ $\sqrt{a^{2} - x^{2}}$
↓
Plug in known values.
$=$ $\sqrt{{(44.2 c m)}^{2} - {(31.7 c m)}^{2}}$
↓
square
$=$ $\sqrt{1953.64 c m^{2} - 1004.89 c m^{2}}$
↓
subtract
$=$ $\sqrt{948.75 c m^{2}}$
↓
take the root
$=$ $30.8 c m$
Compute $α$ usig the sine.
$\sin (α)$ $=$ $\frac{h}{b}$
↓
Plug in known values.
$=$ $\frac{30.8 c m}{44.2 c m}$
↓
Compute $α$ using a calculator.
$α$ $=$ $44.2 ° = β$
↓
So we have a right triangle with $α = β$
$=$
Since the base angles in an isosceles triangle are equal, and all interior angles total $180^{\circ}$ , you may calculate $γ$ .
$γ$ $=$ $180 ° - 2 \cdot 44.2 °$
↓
multiply and subtract
$=$ $91.6 °$
$\Rightarrow$ $h = 30.8 c m; α = β = {44.2}^{\circ}; γ = {91.6}^{\circ}$
Attention: The triangle $A B C$ is not a right triangle because no angle is $90 °$ .
a=114.5m
$α$ =32.3°

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Given: $a = b = 114,5 m$ $α = β$ =32.3°
Required: $c$ , $h$ , $γ$
Draw a sketch for an overview.
Since the base angles in an isosceles triangle are equal, and all interior angles total $180^{\circ}$ , you may calculate $γ$ .
$γ$ $=$ $180 ° - 2 \cdot 32.3 °$
$=$ $115.4 °$
$h$ can be calculated using the sine.
$\sin (α)$ $=$ $\frac{h}{b}$
↓
solve for $h$ and plug in values.
$h$ $=$ $114.5 m \cdot \sin (32.3 °)$
↓
multiply
$=$ $61.2 m$
$x$ is computed using the Pythagorean theorem within the triangle $△ D B C$ .
$x$ $=$ $\sqrt{a^{2} - h^{2}}$
↓
Plug in known values.
$=$ $\sqrt{{(114.5 m)}^{2} - {(61.2 m)}^{2}}$
↓
square
$=$ $\sqrt{13110.25 m^{2} - 3745.44 m^{2}}$
↓
subtract
$=$ $\sqrt{9364.81 m^{2}}$
↓
take the root
$x$ $=$ $96.8 c m$
$c$ can be obtained by doubling the length of $x$ and then cutting the height $h$ of $c$ in half, such that $2$ equally long pieces $x$ are created.
$c = 2 \cdot 96.8 m = 193.6 m$
$\Rightarrow$ $h = 61.2 m; c = 193.6 m; γ = {115.4}^{\circ}$
c=35.4cm
$β$ =43.9°

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Given: c=35.4cm $β = α$ =43.9°
Required: a, b, h, $γ$ , x
Draw a sketch for an overview.
Since the base angles in an isosceles triangle are equal, and all interior angles total $180^{\circ}$ , you may calculate $γ$ .
$γ$ $=$ $180 ° - 2 \cdot 43.9 °$
$=$ $92.2 °$
x is computed by cutting c into half.
$x$ $=$ $\frac{35.4 c m}{2}$
$=$ $17.7 c m$
a can be calculated using the cosine.
$\cos (β)$ $=$ $\frac{x}{a}$
↓
Solve for $a$ and plug in values.
$a$ $=$ $\frac{17.7 c m}{\cos (43.9 °)}$
↓
divide
$a$ $=$ $24.6 c m$
h can be calculated using the tangent.
$\tan (β)$ $=$ $\frac{h}{x}$
↓
Solve for $h$ and plug in values.
$h$ $=$ $17.7 c m \cdot \tan (43.9 °)$
↓
multiply
$h$ $=$ $17.0 c m$
$\Rightarrow$ $α = {43.9}^{\circ}; γ = {92.2}^{\circ}; a = b = 24.6 c m; h = 17.0 c m$
h=14.8cm
$α = β =$ 28.3°

For this task you need the following basic knowledge: Sine, Cosine and Tangent
Given.: $h = 14.8 c m$ ; $α = β = {28.3}^{\circ}$
Required.: $β, γ, c, b, a$
Draw a sketch for an overview.
Since the base angles (here: $α$ and $β$ ) in an isosceles triangle are equal, and all the interior angles add up to $180^{\circ}$ (i.e. $α + β + γ = 180^{\circ}$ ), you can calculate $γ$ directly with this information.
$γ$ $=$ $180 ° - 2 \cdot 28.3 °$
$=$ $123.4 °$
x can be calculated using the tangent.
$\tan (β)$ $=$ $\frac{h}{x}$
↓
Solve for $x$ and plug in values.
$x$ $=$ $\frac{14.8 c m}{\tan (28.3 °)}$
↓
divide
$x$ $=$ $27.5 c m$
↓
You get $c$ by doubling side $x$ .
$c$ $=$ $2 \cdot 27.5 c m$
$=$ $55 c m$
$b$ can be calculated using the sine.
$\sin (α)$ $=$ $\frac{h}{b}$
↓
Solve for $b$ and plug in values.
$b$ $=$ $\frac{14.8 c m}{\sin (28.3 °)}$
↓
divide
$=$ $31.2 c m$
Since this is an isosceles triangle, the side length $a$ is just equal to the side length $b$ .
$\Rightarrow$ $γ = {123.4}^{\circ}; c = 55 c m; a = b = 31.2 c m$

a=146.4m

h=58.4m

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$h$	$=$	$\sqrt{a^{2} - x^{2}}$
	↓	Plug in known values.
	$=$	$\sqrt{{(44.2 c m)}^{2} - {(31.7 c m)}^{2}}$
	↓	square
	$=$	$\sqrt{1953.64 c m^{2} - 1004.89 c m^{2}}$
	↓	subtract
	$=$	$\sqrt{948.75 c m^{2}}$
	↓	take the root
	$=$	$30.8 c m$

$\sin (α)$	$=$	$\frac{h}{b}$
	↓	Plug in known values.
	$=$	$\frac{30.8 c m}{44.2 c m}$
	↓	Compute $α$ using a calculator.
$α$	$=$	$44.2 ° = β$
	↓	So we have a right triangle with $α = β$
	$=$

$γ$	$=$	$180 ° - 2 \cdot 44.2 °$
	↓	multiply and subtract
	$=$	$91.6 °$

$\sin (α)$	$=$	$\frac{h}{b}$
	↓	solve for $h$ and plug in values.
$h$	$=$	$114.5 m \cdot \sin (32.3 °)$
	↓	multiply
	$=$	$61.2 m$

$x$	$=$	$\sqrt{a^{2} - h^{2}}$
	↓	Plug in known values.
	$=$	$\sqrt{{(114.5 m)}^{2} - {(61.2 m)}^{2}}$
	↓	square
	$=$	$\sqrt{13110.25 m^{2} - 3745.44 m^{2}}$
	↓	subtract
	$=$	$\sqrt{9364.81 m^{2}}$
	↓	take the root
$x$	$=$	$96.8 c m$

$\cos (β)$	$=$	$\frac{x}{a}$
	↓	Solve for $a$ and plug in values.
$a$	$=$	$\frac{17.7 c m}{\cos (43.9 °)}$
	↓	divide
$a$	$=$	$24.6 c m$

$\tan (β)$	$=$	$\frac{h}{x}$
	↓	Solve for $h$ and plug in values.
$h$	$=$	$17.7 c m \cdot \tan (43.9 °)$
	↓	multiply
$h$	$=$	$17.0 c m$

$\tan (β)$	$=$	$\frac{h}{x}$
	↓	Solve for $x$ and plug in values.
$x$	$=$	$\frac{14.8 c m}{\tan (28.3 °)}$
	↓	divide
$x$	$=$	$27.5 c m$
	↓	You get $c$ by doubling side $x$ .
$c$	$=$	$2 \cdot 27.5 c m$
	$=$	$55 c m$

$\sin (α)$	$=$	$\frac{h}{b}$
	↓	Solve for $b$ and plug in values.
$b$	$=$	$\frac{14.8 c m}{\sin (28.3 °)}$
	↓	divide
	$=$	$31.2 c m$

$\sin (β)$	$=$	$\frac{h}{a}$
	↓	Plug in values and compute $α$
$β$	$=$	$23.5 °$