Geometry Course/Polygon/Triangle/Right Triangle

Right Triangle

Right Triangle is a triangle that has one angle equal to 90^o and the sum of the other two angles equal to 90^o

Pythagoras's Theorem

The three sides of the right triangle relate to each other as shown

(Hypotenuse)² = (Adjacent)² + (Opposite)²

$b^{2}=a^{2}+c^{2}$

Trigonometry Triangle

In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known as follows:

• The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
• The opposite side is the side opposite to the angle we are interested in, in this case a.
• The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

Sine

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

$\sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{h}}\,.$

Cosine

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

$\cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{h}}\,.$

Tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case

$\tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{b}}\,.$

The sine, cosine and tangent rules

A triangle with sides of length a, b and c and angles of α, β and γ respectively.

The law of sines, or sine rule, states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is

${\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}.$

This ratio is equal to the diameter of the circumscribed circle of the given triangle. Another interpretation of this theorem is that every triangle with angles $\alpha$, $\beta$ and $\gamma$ is similar to a triangle with side lengths equal to $\sin \alpha$, $\sin \beta$ and $\sin \gamma$. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be $\sin \alpha$, $\sin \beta$ and $\sin \gamma$. The side whose length is $\sin \alpha$ is opposite to the angle whose measure is $\alpha$, etc.

The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side. As per the law:

For a triangle with length of sides $a$, $b$, $c$ and angles of $\alpha$, $\beta$, $\gamma$ respectively, given two known lengths of a triangle $a$ and $b$, and the angle between the two known sides $\gamma$ (or the angle opposite to the unknown side $c$), to calculate the third side $c$, the following formula can be used:

$c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma )$

$b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta )$

$a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha )$

If the lengths of all three sides of any triangle are known the three angles can be calculated:

$\alpha =\arccos \left({\frac {b^{2}+c^{2}-a^{2}}{2bc}}\right)$

$\beta =\arccos \left({\frac {a^{2}+c^{2}-b^{2}}{2ac}}\right)$

$\gamma =\arccos \left({\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)$

The law of tangents or tangent rule, is less known than the other two. It states that:

${\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.$

It is not used very often, but can be used to find a side or an angle when you know two sides and an angle or two angles and a side.