A ratio is a statement of a mathematical relationship comparing two quantities, often represented as a fraction. If we consider an angle $\theta$θ in a right-angled triangle, we can construct various ratios to compare the lengths of the sides. Special relationships that exist in right-angled triangles are called trigonometric ratios.
Through investigation we can see that there is a definite relationship between the angles in a right-angled triangle and the ratio of sides. We can use these trigonometric ratios to find unknown angles and sides of a triangle.
There are 3 basic trigonometric ratios that relate sides and angles together. They have the special names of tangent, sine and cosine. The names of the relationships date back to 499AD, with ties to Latin and Sanskrit. Actually the word sine is thought to be derived from a translation gone wrong! Regardless the names for these relationships have stuck.
We often shorten the names tangent, sine and cosine to tan, sin and cos respectively.
$\sin\theta$sinθ = $\frac{Opposite}{Hypotenuse}$OppositeHypotenuse = $\frac{b}{c}$bc
$\cos\theta$cosθ = $\frac{Adjacent}{Hypotenuse}$AdjacentHypotenuse = $\frac{a}{c}$ac
$\tan\theta$tanθ = $\frac{Opposite}{Adjacent}$OppositeAdjacent = $\frac{b}{a}$ba
Here is a picture of the above relationships, and for some people the mnemonic of SOHCAHTOA at the bottom is helpful to remember the sides that apply to the different ratios of sine, cosine and tangent.
For a particular angle in a right-angled triangle, the sin ratio compares the length of the side opposite the angle to the length of the hypotenuse.
The cos ratio compares the length of the side adjacent the angle to the length of the hypotenuse.
Consider the following triangles. $\triangle MOP$△MOP is an enlargement of $\triangle MNQ$△MNQ. This means the angles REMAIN THE SAME between the two triangles.
Consider angle A | In $\triangle MNQ$△MNQ | In $\triangle MOP$△MOP |
---|---|---|
Adjacent | 4 | 8 |
Opposite | 3 | 6 |
Hypotenuse | 5 | 10 |
sin A ($\frac{opposite}{hypotenuse}$oppositehypotenuse) | $\frac{3}{5}$35 | $\frac{6}{10}=\frac{3}{5}$610=35 |
cos A ($\frac{adjacent}{hypotenuse}$adjacenthypotenuse) | $\frac{4}{5}$45 | $\frac{8}{10}=\frac{4}{5}$810=45 |
Overlapping the two triangles:
So even though the side lengths in the two triangles are different:
- the cos ratio of angle A ( $\frac{adjacent}{hypotenuse}$adjacenthypotenuse) remains exactly the same, and
- the sin ratio of angle A ( $\frac{adjacent}{hypotenuse}$adjacenthypotenuse) remains exactly the same.
Later on we will be able to use this property (that every angle has a fixed sin and cos ratio) to find unknown angles.
This applet will let you explore different trigonometric ratios, particularly which sides are needed for a particular ratio. If you would like to see this applet in action, watch this video.
Let's have a look at just one more special relationship.
If we know the sine and cosine ratios for a particular angle,
$\sin\theta=\frac{Opposite}{Hypotenuse}$sinθ=OppositeHypotenuse
$\cos\theta=\frac{Adjacent}{Hypotenuse}$cosθ=AdjacentHypotenuse
Then we can construct a new relationship for $\text{sine }\div\text{cosine }$sine ÷cosine
$\frac{\sin\theta}{\cos\theta}$sinθcosθ | $=$= | $\frac{\left(\frac{Opposite}{Hypotenuse}\right)}{\left(\frac{Adjacent}{Hypotenuse}\right)}$(OppositeHypotenuse)(AdjacentHypotenuse) |
$=$= | $\frac{Opposite}{Hypotenuse}\times\frac{Hypotenuse}{Adjacent}$OppositeHypotenuse×HypotenuseAdjacent | |
$=$= | $\frac{Opposite}{Adjacent}$OppositeAdjacent | |
$=$= | $\tan\theta$tanθ |
Algebraically we have just shown that $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ
That is, the tangent ratio of an angle is the same as dividing its sine ratio by its cosine ratio.
Let's have a look at some worked examples.
Write down the ratio of the sides represented by $\sin\theta$sinθ.
Write down the ratio represented by $\cos\theta$cosθ.
Evaluate $\cos\theta$cosθ within $\triangle ABC$△ABC.
Consider the following triangle.
Find the value of $x$x.
Hence find the value of $\sin\theta$sinθ.
Hence find the value of $\cos\theta$cosθ.
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