Naming the sides of a right-angled triangle
When using Pythagoras' theorem we saw that the longest side of a right-angled triangle is called the hypotenuse.
If we have another angle indicated (like $x$x in the diagram below) then we can also label the other two sides with special names according to their position in relation to this angle as follows:
Opposite Side - is the name given to the side opposite the angle in question
Adjacent Side - is the name given to the side adjacent (next to) the angle in question.
Have a look at these triangles below and the naming of the sides in relation to the angle shaded red. Note how the sides adjacent, opposite and hypotenuse are also abbreviated to $A$A, $O$O and $H$H.
Practice questions
Question 1
Which of the following is the opposite side to angle $\theta$θ?
A right-angled triangle has vertices labeled in a counterclockwise direction: vertex $A$A is positioned at the top, vertex $B$B at the bottom left, and vertex $C$C at the bottom right. At vertex $A$A, $\angle BAC$∠BAC is labeled as $\alpha$α. At vertex $B$B, $\angle ABC$∠ABC is labeled as $\theta$θ. At vertex $C$C , $\angle ACB$∠ACB is the right angle denoted by a small square. The hypotenuse is the side $overline(AB)$overline(AB). Side $overline(BC)$overline(BC) is opposite to angle $\alpha$α. $Sideoverline(BC)$Sideoverline(BC) is adjacent to angle $\theta$θ. Side $overline(AC)$overline(AC) is opposite to angle $\theta$θ. $Sideoverline(AC)$Sideoverline(AC) is adjacent to angle $\alpha$α. Side $overline(AC)$overline(AC) is not adjacent to angle $theta$theta.
$AB$AB
A$BC$BC
B$AC$AC
C
Question 2
Which of the following is the adjacent side to angle $\theta$θ?
$AB$AB
A$BC$BC
B$AC$AC
C
Question 3
A driver glances up at the top of a building.
True or false: According to the angle A, the height of the building is the opposite side.
True
AFalse
BTrue or false: According to the angle A, the distance from the driver to the building would be the opposite side.
True
AFalse
B
The ratio of sides of a right-angled triangle
A ratio is a statement of a mathematical relationship between two objects, often represented as a fraction. If we consider and two sides of a right-angled triangle with respect to a given internal angle, the ratio of the two sides has a special relationship to the angle. Consider the following three ratios formed according to their position in relation to the angle $\theta$θ:
For triangles with the same internal angles, the ratios of the sides remain constant. For example, consider the three right-angled triangles below where the other two angles are $45^\circ$45°. This means they are also isosceles triangles, with two equal sides.
As the triangles get larger, the side lengths increase, but the ratio of $\frac{\text{Opposite }}{\text{Adjacent }}$Opposite Adjacent from the $45^\circ$45° angle always stays the same. In each triangle, the ratio is $\frac{1}{1}$11, $\frac{2}{2}$22 and $\frac{3}{3}$33 and all of these ratios simplify to $1$1. We can explore these constant ratios for any angle in a right-angled triangle, not just $45^\circ$45°.
Worked examples
Example 1
Considering the angle $\theta$θ, what is the value of the ratio $\frac{Adjacent}{Hypotenuse}$AdjacentHypotenuse ?
Think: First we need to identify which sides are the adjacent and hypotenuse with respect to angle theta. We can see that $BA$BA is the hypotenuse, $AC$AC is the opposite side and $BC$BC is the adjacent.
Do: $\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{BC}{AB}=\frac{5}{13}$AdjacentHypotenuse=BCAB=513
Practice questions
Question 4
Question 5
Consider the angle $\theta$θ.
What is the value of the ratio $\frac{Opposite}{Adjacent}$OppositeAdjacent?
Express your answer as a fraction.
Trigonometric ratios
The three ratios mentioned above are related to their respective angles by trigonometric ratios. The three ratios are Sine, Cosine and Tangent, shortened to become sin, cos and tan respectively. The trigonometric ratios enable us to determine the ratio of two sides of a right-angled triangle given an internal angle or find an angle given the ratio of two sides of a right-angled triangle. The three trigonometric ratios are defined as follows:
$\sin\left(\theta\right)=\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{O}{H}$sin(θ)=OppositeHypotenuse=OH
$\cos\left(\theta\right)=\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{A}{H}$cos(θ)=AdjacentHypotenuse=AH
$\tan\left(\theta\right)=\frac{\text{Opposite}}{\text{Adjacent}}=\frac{O}{A}$tan(θ)=OppositeAdjacent=OA
The mnemonic of SOH CAH TOA is helpful to remember the sides that apply to the different ratios of sine, cosine and tangent.
Every angle has a fixed $\sin$sin, $\cos$cos, and $\tan$tan ratio, so they are programmed into our calculators.
For example, we can use our calculators to find the following values:
- $\sin30^\circ=0.5$sin30°=0.5 (The side opposite $30^\circ$30° is half the length of the hypotenuse)
- $\cos25^\circ=0.9$cos25°=0.9 (The side adjacent to $25^\circ$25° is $0.9$0.9 times the length of the hypotenuse)
- $\tan82^\circ=7.1$tan82°=7.1 (The side opposite $82^\circ$82° is about $7$7 times the length of the adjacent side)
Having these trigonometric ratios allows us to solve for other unknowns. For example, we can calculate the unknown sides and angles of right-angled triangles given sufficient information, something we will look at in the next lesson.
Practice questions
Question 6
Consider the triangle in the figure. If $\sin\theta=\frac{4}{5}$sinθ=45:
Which angle is represented by $\theta$θ?
$\angle BAC$∠BAC
A$\angle BCA$∠BCA
B$\angle ABC$∠ABC
CFind the value of $\cos\theta$cosθ. Express your answer as a simplified fraction.
Find the value of $\tan\theta$tanθ. Express your answer as a simplified fraction.
Question 7
Evaluate $\tan\theta$tanθ within $\triangle ABC$△ABC. Express your answer as a simplified fraction.
Question 8
Consider the triangle in the figure.
If $\cos\theta=\frac{6}{10}$cosθ=610:
Which angle is represented by $\theta$θ?
$\angle BCA$∠BCA
A$\angle BAC$∠BAC
B$\angle ABC$∠ABC
CFind the numerical value of $\sin\theta$sinθ. Express your answer as a simplified fraction.
Find the numerical value of $\tan\theta$tanθ. Express your answer as a simplified fraction.