Trigonometry

Ontario 10 Applied (MFM2P)

Identify ratio of sides in right-angled triangles

Lesson

In the above right-angled triangle $c$`c` is the the hypotenuse and $\angle RPQ$∠`R``P``Q` is $90^\circ$90°

Now if we call $\angle PQR=\theta$∠`P``Q``R`=`θ` and we can identify the opposite and adjacent sides with respect to that angle, (we did this here, if you need to refresh). So $b$`b` is the opposite side, and $a$`a` is the adjacent side.

A ratio is a statement of a mathematical relationship between two objects, often represented as a fraction. Various ratios of the following can be constructed from the right-angled triangle with respect to angle $\theta$`θ`.

$\frac{Opposite}{Adjacent}=\frac{b}{a}$`O``p``p``o``s``i``t``e``A``d``j``a``c``e``n``t`=`b``a`

$\frac{Adjacent}{Hypotenuse}=\frac{a}{c}$`A``d``j``a``c``e``n``t``H``y``p``o``t``e``n``u``s``e`=`a``c`

$\frac{Opposite}{Hypotenuse}=\frac{b}{c}$`O``p``p``o``s``i``t``e``H``y``p``o``t``e``n``u``s``e`=`b``c`

Considering the angle $\theta$`θ`, what is the value of the ratio $\frac{Adjacent}{Hypotenuse}$`A``d``j``a``c``e``n``t``H``y``p``o``t``e``n``u``s``e` ?

**Think**: First we need to identify which sides are the adjacent and hypotenuse with respect to angle theta. I can see that $BA$`B``A` is the hypotenuse, $AC$`A``C` is the opposite side and $BC$`B``C` is the adjacent.

**Do**: $\frac{Adjacent}{Hypotenuse}$`A``d``j``a``c``e``n``t``H``y``p``o``t``e``n``u``s``e` = $\frac{BC}{AB}=\frac{5}{13}$`B``C``A``B`=513

Consider the angle $\theta$`θ`.

What is the value of the ratio $\frac{Opposite}{Hypotenuse}$`O``p``p``o``s``i``t``e``H``y``p``o``t``e``n``u``s``e`?

Consider the angle $\theta$`θ`.

What is the value of the ratio $\frac{Opposite}{Adjacent}$`O``p``p``o``s``i``t``e``A``d``j``a``c``e``n``t`?

Determine, through investigation, the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios