Exercises: Sine, Cosine and Tangent on a right triangle

Calculate the missing sides and angles of the isosceles triangle $ABC$ with $a=b$ .

a = 44.2cm

c = 63.4cm

a = 114.5m

$\alpha$ = 32.3°

For this task you need the following basic knowledge: Triangle

Calculate the side $c$ of the triangle:

Use the cosine to calculate the base of the triangle,

$\displaystyle \cos\left(\alpha\right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{b}$	$\displaystyle \cdot b$
$\displaystyle \cos\left(32.3°\right)\cdot114.5m$	$=$	$\displaystyle \frac{1}{2}c$	$\displaystyle \cdot2$
$\displaystyle 2\cdot\left(\cos\left(32.3°\right)\cdot114.5m\right)$	$=$	$\displaystyle c$
$\displaystyle c$	$=$	$\displaystyle 193.565$

Calculate the height of the triangle:

You may use the sine to do this.

$\displaystyle \sin\left(\alpha\right)$	$=$	$\displaystyle \frac{h_c}{b}$	$\displaystyle \cdot b$
$\displaystyle \sin\left(32.3°\right)\cdot114.5m$	$=$	$\displaystyle h_c$
$\displaystyle h_c$	$=$	$\displaystyle 61.183m$
	↓	Calculate the two missing angles using sine or cosine
	$=$

$\displaystyle \cos\left(\beta\right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{a}$
$\displaystyle \beta$	$=$	$\displaystyle 32.3°$

$\displaystyle \sin\left(\beta\right)$	$=$	$\displaystyle \frac{h}{a}$
$\displaystyle \beta$	$=$	$\displaystyle 32.3°$

$\displaystyle \gamma$	$=$	$\displaystyle 180°-2\cdot32.3°$
$\displaystyle \gamma$	$=$	$\displaystyle 115.4°$

alternatively:

$\displaystyle \cos\left(\frac{1}{2}\gamma\right)$	$=$	$\displaystyle \frac{h_c}{b}$
$\displaystyle \frac{1}{2}\gamma$	$=$	$\displaystyle 57.25°$
$\displaystyle \gamma$	$=$	$\displaystyle 115.4°$
$\displaystyle \sin\left(\frac{1}{2}\gamma\right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{b}$
$\displaystyle \frac{1}{2}\gamma$	$=$	$\displaystyle 57.25°$	$\displaystyle \cdot2$
$\displaystyle \gamma$	$=$	$\displaystyle 115.4$

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c = 35.4cm

$\beta$ = 43.9°

For this task you need the following basic knowledge: Triangle

Calculate the sides $a$ and $b$ of the triangle.

Use the cosine to calculate the two catheti,

$\displaystyle \cos\left(\beta\right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{a}$	$\displaystyle \cdot a:\cos\left(\beta\right)$
$\displaystyle a$	$=$	$\displaystyle \frac{\frac{1}{2}\cdot35.4cm}{\cos\left(43.9°\right)}$
$\displaystyle a$	$=$	$\displaystyle 24.565cm$
	$=$	$\displaystyle b$

Calculate the height of the triangle:

You may use the sine to do this.

$\displaystyle \sin\left(\beta\right)$	$=$	$\displaystyle \frac{h_c}{a}$	$\displaystyle \cdot a$
$\displaystyle h_c$	$=$	$\displaystyle \sin\left(43.9°\right)\cdot24.565cm$
$\displaystyle h_c$	$=$	$\displaystyle 17.033cm$
	↓	Calculate the two missing angles with sine or cosine .
	$=$

$\displaystyle \cos\left(\alpha\ \right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{b}$
$\displaystyle \alpha$	$=$	$\displaystyle 43.9°$

or:

$\displaystyle \sin\left(\alpha\right)$	$=$	$\displaystyle \frac{h_c}{b}$
$\displaystyle \alpha$	$=$	$\displaystyle 43.9°$
	↓	Add the two angles and subtract them from the sum of the angles in the triangle to calculate the last angle.
	$=$

$\displaystyle \gamma$	$=$	$\displaystyle 180°-2\cdot43.9°$
	$=$	$\displaystyle 92.2°$

alternatively:

$\displaystyle \cos\left(\frac{1}{2}\gamma\right)$	$=$	$\displaystyle \frac{h_c}{b}$
$\displaystyle \frac{1}{2}\gamma$	$=$	$\displaystyle 46.1°$	$\displaystyle \cdot2$
$\displaystyle \gamma$	$=$	$\displaystyle 92.2°$
$\displaystyle \sin\left(\frac{1}{2}\gamma\right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{b}$
$\displaystyle \frac{1}{2}\gamma$	$=$	$\displaystyle 46.1°$	$\displaystyle \cdot2$
$\displaystyle \gamma$	$=$	$\displaystyle 92.2°$

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$h_c$ = 14.8cm

$\alpha$ = 28.3°

For this task you need the following basic knowledge: Triangle

Calculate the sides a and b of the triangle:

You may use the sine to do this.

$\displaystyle \sin\left(\alpha\right)$	$=$	$\displaystyle \frac{h_c}{b}$	$\displaystyle \cdot b:\sin\left(\alpha\right)$
$\displaystyle b$	$=$	$\displaystyle \frac{14.8\ cm}{\sin\left(28.3°\right)}$
$\displaystyle b$	$=$	$\displaystyle 31.218$
	$=$	$\displaystyle a$

Calculate the side $c$ of the triangle:

You may use the cosine to do this.

$\displaystyle \cos\left(\alpha\right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{b}$	$\displaystyle \cdot b$
$\displaystyle \cos\left(28.3°\right)\cdot31.218cm$	$=$	$\displaystyle \frac{1}{2}c$	$\displaystyle \cdot2$
$\displaystyle c$	$=$	$\displaystyle 54.973cm$
	↓	Calculate the missing angles using cosine or sine,
	$=$

$\displaystyle \cos\left(\beta\right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{a}$
$\displaystyle \beta$	$=$	$\displaystyle 28.3°$

or:

$\displaystyle \sin\left(\beta\right)$	$=$	$\displaystyle \frac{h_c}{a}$
$\displaystyle \beta$	$=$	$\displaystyle 28.3°$
	↓	Add the two angles and subtract them from the sum of the angles in the triangle to work out the last angle.
	$=$

$\displaystyle \gamma$	$=$	$\displaystyle 180°-2\cdot28{,}3°$
$\displaystyle \gamma$	$=$	$\displaystyle 123{,}4°$

alternatively:

$\displaystyle \cos\left(\frac{1}{2}\gamma\ \right)$	$=$	$\displaystyle \frac{h_c}{b}$
$\displaystyle \frac{1}{2}\gamma$	$=$	$\displaystyle 61{,}7°$	$\displaystyle \cdot2$
$\displaystyle \gamma$	$=$	$\displaystyle 123{,}4°$

or:

$\displaystyle \sin\left(\frac{1}{2}\gamma\right)$	$=$	$\displaystyle \frac{\frac{1}{2}c}{b}$
$\displaystyle \frac{1}{2}\gamma$	$=$	$\displaystyle 61{,}7°$	$\displaystyle \cdot2$
$\displaystyle \gamma$	$=$	$\displaystyle 123{,}4°$

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a = 146.4m

$h_c$ = 58.4m

$\displaystyle \cos\left(\frac{1}{2}\gamma\right)$	$=$	$\displaystyle \frac{{\displaystyle\frac12}c}a$
$\displaystyle \frac{1}{2}\gamma$	$≈$	$\displaystyle 45.8^°$
$\displaystyle \gamma$	$≈$	$\displaystyle 45.8^{\circ}\cdot2=91.6^{\circ}$

$\displaystyle \sin\left(\beta\right)$	$=$	$\displaystyle \frac{h_c}{\alpha}$
$\displaystyle \beta$	$≈$	$\displaystyle 44.177°$

$\displaystyle \gamma$	$=$	$\displaystyle 180°-2\cdot44.177°.$
$\displaystyle \gamma$	$=$	$\displaystyle 91.646°$

$\displaystyle \gamma$	$=$	$\displaystyle 180^{\circ}-2\cdot23.51^{\circ}$
	$=$	$\displaystyle 132.98^{\circ}$