Exercises: Sine, Cosine and Tangent on a right triangle

Calculate the missing sides and angles of the isosceles triangle $A B C$ with $a = b$ .

a = 44.2cm
c = 63.4cm

Calculate the height of the triangle.
Apply the Pythagorean theorem to calculate the height.
$h_{c} = \sqrt{b^{2} - {(\frac{1}{2} c)}^{2}} \approx 30.802 c m$
Now calculate $α$ , $β$ and $γ$ using sine or cosine.
$\sin (α)$ $=$ $\frac{h_{c}}{b}$
$α$ $\approx$ $44,177 ° .$
or:
$\cos (α)$ $=$ $\frac{\frac{1}{2} c}{b}$
$α$ $=$ $44.177 °$
$\sin (β)$ $=$ $\frac{h_{c}}{α}$
$β$ $\approx$ $44.177 °$
$\cos (β)$ $=$ $\frac{\frac{1}{2} c}{a}$
$β$ $=$ $44.177 °$
Add the two angles and subtract them from the sum of the angles in the triangle.
$γ$ $=$ $180 ° - 2 \cdot 44.177 ° .$
$γ$ $=$ $91.646 °$
alternatively:
$\cos (\frac{1}{2} γ)$ $=$ $\frac{\frac{1}{2} c}{a}$
$\frac{1}{2} γ$ $\approx$ ${45.8}^{°}$
$γ$ $\approx$ ${45.8}^{\circ} \cdot 2 = {91.6}^{\circ}$
or:
$\sin (\frac{1}{2} γ)$ $=$ $\frac{h_{c}}{a}$
$\frac{1}{2} γ$ $=$ $45.8 °$
$γ$ $=$ $2 \cdot 45.8 ° = 91.6 °$
a = 114.5m
$α$ = 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,
$\cos (α)$ $=$ $\frac{\frac{1}{2} c}{b}$ $\cdot b$
$\cos (32.3 °) \cdot 114.5 m$ $=$ $\frac{1}{2} c$ $\cdot 2$
$2 \cdot (\cos (32.3 °) \cdot 114.5 m)$ $=$ $c$
$c$ $=$ $193.565$
Calculate the height of the triangle:
You may use the sine to do this.
$\sin (α)$ $=$ $\frac{h_{c}}{b}$ $\cdot b$
$\sin (32.3 °) \cdot 114.5 m$ $=$ $h_{c}$
$h_{c}$ $=$ $61.183 m$
↓
Calculate the two missing angles using sine or cosine
$=$
$\cos (β)$ $=$ $\frac{\frac{1}{2} c}{a}$
$β$ $=$ $32.3 °$
$\sin (β)$ $=$ $\frac{h}{a}$
$β$ $=$ $32.3 °$
$γ$ $=$ $180 ° - 2 \cdot 32.3 °$
$γ$ $=$ $115.4 °$
alternatively:
$\cos (\frac{1}{2} γ)$ $=$ $\frac{h_{c}}{b}$
$\frac{1}{2} γ$ $=$ $57.25 °$
$γ$ $=$ $115.4 °$
$\sin (\frac{1}{2} γ)$ $=$ $\frac{\frac{1}{2} c}{b}$
$\frac{1}{2} γ$ $=$ $57.25 °$ $\cdot 2$
$γ$ $=$ $115.4$
c = 35.4cm
$β$ = 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,
$\cos (β)$ $=$ $\frac{\frac{1}{2} c}{a}$ $\cdot a : \cos (β)$
$a$ $=$ $\frac{\frac{1}{2} \cdot 35.4 c m}{\cos (43.9 °)}$
$a$ $=$ $24.565 c m$
$=$ $b$
Calculate the height of the triangle:
You may use the sine to do this.
$\sin (β)$ $=$ $\frac{h_{c}}{a}$ $\cdot a$
$h_{c}$ $=$ $\sin (43.9 °) \cdot 24.565 c m$
$h_{c}$ $=$ $17.033 c m$
↓
Calculate the two missing angles with sine or cosine .
$=$
$\cos (α)$ $=$ $\frac{\frac{1}{2} c}{b}$
$α$ $=$ $43.9 °$
or:
$\sin (α)$ $=$ $\frac{h_{c}}{b}$
$α$ $=$ $43.9 °$
↓
Add the two angles and subtract them from the sum of the angles in the triangle to calculate the last angle.
$=$
$γ$ $=$ $180 ° - 2 \cdot 43.9 °$
$=$ $92.2 °$
alternatively:
$\cos (\frac{1}{2} γ)$ $=$ $\frac{h_{c}}{b}$
$\frac{1}{2} γ$ $=$ $46.1 °$ $\cdot 2$
$γ$ $=$ $92.2 °$
$\sin (\frac{1}{2} γ)$ $=$ $\frac{\frac{1}{2} c}{b}$
$\frac{1}{2} γ$ $=$ $46.1 °$ $\cdot 2$
$γ$ $=$ $92.2 °$
$h_{c}$ = 14.8cm
$α$ = 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.
$\sin (α)$ $=$ $\frac{h_{c}}{b}$ $\cdot b : \sin (α)$
$b$ $=$ $\frac{14.8 c m}{\sin (28.3 °)}$
$b$ $=$ $31.218$
$=$ $a$
Calculate the side $c$ of the triangle:
You may use the cosine to do this.
$\cos (α)$ $=$ $\frac{\frac{1}{2} c}{b}$ $\cdot b$
$\cos (28.3 °) \cdot 31.218 c m$ $=$ $\frac{1}{2} c$ $\cdot 2$
$c$ $=$ $54.973 c m$
↓
Calculate the missing angles using cosine or sine,
$=$
$\cos (β)$ $=$ $\frac{\frac{1}{2} c}{a}$
$β$ $=$ $28.3 °$
or:
$\sin (β)$ $=$ $\frac{h_{c}}{a}$
$β$ $=$ $28.3 °$
↓
Add the two angles and subtract them from the sum of the angles in the triangle to work out the last angle.
$=$
$γ$ $=$ $180 ° - 2 \cdot 28,3 °$
$γ$ $=$ $123,4 °$
alternatively:
$\cos (\frac{1}{2} γ)$ $=$ $\frac{h_{c}}{b}$
$\frac{1}{2} γ$ $=$ $61,7 °$ $\cdot 2$
$γ$ $=$ $123,4 °$
or:
$\sin (\frac{1}{2} γ)$ $=$ $\frac{\frac{1}{2} c}{b}$
$\frac{1}{2} γ$ $=$ $61,7 °$ $\cdot 2$
$γ$ $=$ $123,4 °$
a = 146.4m
$h_{c}$ = 58.4m

For this task you need the following basic knowledge: Triangle
Calculate the angle $α$ .
You may use the sine to do this.
$\sin (α)$ $=$ $\frac{h_{c}}{b}$
$α$ $=$ ${23.51}^{\circ}$
Now calculate the side $c$ using the cosine.
$\cos (α)$ $=$ $\frac{\frac{1}{2} c}{b}$ $\cdot b$
$\frac{1}{2} c$ $=$ $\cos (α) \cdot b$
$\frac{1}{2} c$ $=$ $134.25 m$ $\cdot 2$
$c$ $=$ $268.5 m$
Calculate the remaining angles using sine or cosine.
$\cos (β)$ $=$ $\frac{\frac{1}{2} c}{a}$
$β$ $=$ ${23.51}^{\circ}$
or:
$\sin (β)$ $=$ $\frac{h_{c}}{a}$
$β$ $=$ ${23.51}^{\circ}$
Add the two angles and subtract them from the sum of the angles in the triangle.
$γ$ $=$ $180^{\circ} - 2 \cdot {23.51}^{\circ}$
$=$ ${132.98}^{\circ}$
alternatively:
$\cos (\frac{1}{2} γ)$ $=$ $\frac{h_{c}}{b}$
$\frac{1}{2} γ$ $=$ ${66.49}^{\circ}$ $\cdot 2$
$γ$ $=$ ${132.98}^{\circ}$
or:
$\sin (\frac{1}{2} γ)$ $=$ $\frac{\frac{1}{2} c}{b}$
$\frac{1}{2} γ$ $=$ ${66.49}^{\circ}$ $\cdot 2$
$γ$ $=$ ${132.98}^{\circ}$

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$\cos (\frac{1}{2} γ)$	$=$	$\frac{\frac{1}{2} c}{a}$
$\frac{1}{2} γ$	$\approx$	${45.8}^{°}$
$γ$	$\approx$	${45.8}^{\circ} \cdot 2 = {91.6}^{\circ}$

$\sin (\frac{1}{2} γ)$	$=$	$\frac{h_{c}}{a}$
$\frac{1}{2} γ$	$=$	$45.8 °$
$γ$	$=$	$2 \cdot 45.8 ° = 91.6 °$

$\cos (α)$	$=$	$\frac{\frac{1}{2} c}{b}$	$\cdot b$
$\cos (32.3 °) \cdot 114.5 m$	$=$	$\frac{1}{2} c$	$\cdot 2$
$2 \cdot (\cos (32.3 °) \cdot 114.5 m)$	$=$	$c$
$c$	$=$	$193.565$

$\sin (α)$	$=$	$\frac{h_{c}}{b}$	$\cdot b$
$\sin (32.3 °) \cdot 114.5 m$	$=$	$h_{c}$
$h_{c}$	$=$	$61.183 m$
	↓	Calculate the two missing angles using sine or cosine
	$=$

$\cos (β)$	$=$	$\frac{\frac{1}{2} c}{a}$	$\cdot a : \cos (β)$
$a$	$=$	$\frac{\frac{1}{2} \cdot 35.4 c m}{\cos (43.9 °)}$
$a$	$=$	$24.565 c m$
	$=$	$b$

$\sin (β)$	$=$	$\frac{h_{c}}{a}$	$\cdot a$
$h_{c}$	$=$	$\sin (43.9 °) \cdot 24.565 c m$
$h_{c}$	$=$	$17.033 c m$
	↓	Calculate the two missing angles with sine or cosine .
	$=$

$\sin (α)$	$=$	$\frac{h_{c}}{b}$
$α$	$=$	$43.9 °$
	↓	Add the two angles and subtract them from the sum of the angles in the triangle to calculate the last angle.
	$=$

$\sin (α)$	$=$	$\frac{h_{c}}{b}$	$\cdot b : \sin (α)$
$b$	$=$	$\frac{14.8 c m}{\sin (28.3 °)}$
$b$	$=$	$31.218$
	$=$	$a$

$\sin (β)$	$=$	$\frac{h_{c}}{a}$
$β$	$=$	$28.3 °$
	↓	Add the two angles and subtract them from the sum of the angles in the triangle to work out the last angle.
	$=$

$γ$	$=$	$180^{\circ} - 2 \cdot {23.51}^{\circ}$
	$=$	${132.98}^{\circ}$