16.01 Trigonometric ratios
Trigonometric ratios
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. In a right-angled triangle the ratios of the sides are the trigonometric ratios. Three common trigonometric ratios we use are Sine, Cosine and Tangent, we often shorten these names to sin, cos and tan respectively. They are given by the ratio of sides relative to the given angle $\theta$θ.
| $\sin\left(\theta\right)$sin(θ) | $=$= | $\frac{\text{Opposite}}{\text{Hypotenuse}}$OppositeHypotenuse | $=$= | $\frac{O}{H}$OH |
| $\cos\left(\theta\right)$cos(θ) | $=$= | $\frac{\text{Adjacent}}{\text{Hypotenuse}}$AdjacentHypotenuse | $=$= | $\frac{A}{H}$AH |
| $\tan\left(\theta\right)$tan(θ) | $=$= | $\frac{\text{Opposite}}{\text{Adjacent}}$OppositeAdjacent | $=$= | $\frac{O}{A}$OA |
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.
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For a right-angled triangle with a given angle, say $20^\circ$20°, we know the third angle must be $70^\circ$70° and this would be true for any right-angled triangle with one angle of $20^\circ$20°. Hence, all right-angled triangles with an angle of $20^\circ$20° are similar and will have the same ratio of given sides. The calculator can approximate the ratio very accurately, typing in the calculator $\sin\left(20^\circ\right)$sin(20°), will tell us that the ratio for the opposite side divided by the hypotenuse for any right-angled triangle with an angle $20^\circ$20° is approximately $0.342$0.342. We can use these trigonometric ratios to find unknown sides of a right-angled triangle given an angle or an unknown angle given two sides.
To find the angle using each of these relationships, we can apply the inverse trigonometric function to each side of the equation. This will give us:
- $\theta=\sin^{-1}\left(\frac{\text{Opposite }}{\text{Hypotenuse }}\right)$θ=sin−1(Opposite Hypotenuse )
- $\theta=\cos^{-1}\left(\frac{\text{Adjacent }}{\text{Hypotenuse }}\right)$θ=cos−1(Adjacent Hypotenuse )
- $\theta=\tan^{-1}\left(\frac{\text{Opposite }}{\text{Adjacent }}\right)$θ=tan−1(Opposite Adjacent )
Finding a side
Based on where the angle is in the triangle and which pair of sides we are working with, we can choose one of the trigonometric ratios to describe the relationship between those values.
We can then rearrange that ratio to make our unknown value the subject of an equation and then evaluate to find its value.
Worked example
Find the value of $x$x.
Think: With respect to the given angle, the side of length $5$5 is adjacent and the side of length $x$x is opposite. This means that we should choose the trigonometric ratio $\tan\theta=\frac{\text{Opposite }}{\text{Adjacent }}$tanθ=Opposite Adjacent to relate our given values.
Do: Substituting our given values into the trigonometric ratio gives us:
$\tan38^\circ=\frac{x}{5}$tan38°=x5
We can multiply both sides of the equation by $5$5 to make $x$x the subject and then evaluate to find its value (rounded to two decimal places):
$x=5\tan38^\circ$x=5tan38°$=$=$3.91$3.91
Reflect: After identifying which sides we were working with, we chose the trigonometric ratio that matched those sides. We then solved the equation to find our unknown side length.
Finding an angle
Based on where the angle is in the triangle and which pair of sides we are given, we can choose one of the trigonometric ratios to describe the relationship between those values.
We can then rearrange that ratio (or choose the corresponding inverse ratio) to make our unknown angle the subject of an equation and then solve for it.
Worked example
Find the value of $x$x.
Think: With respect to the angle $x$x, the side of length $4$4 is opposite and the side of length $5$5 is adjacent. This means that we can use the inverse trigonometric ratio $\theta=\tan^{-1}\left(\frac{\text{Opposite }}{\text{Adjacent }}\right)$θ=tan−1(Opposite Adjacent ).
Do: Substituting our values into the inverse trigonometric ratio gives us:
$x=\tan^{-1}\frac{4}{5}$x=tan−145
Evaluating $x$x (and rounding to two decimal places) gives us:
$x=38.66$x=38.66
Reflect: After identifying which sides we were given, we chose the inverse trigonometric ratio that matched those sides. We then solved the equation to find our unknown angle size.
Practice questions
Question 1
Consider the angle $\theta$θ.
What is the value of the ratio $\frac{\text{Opposite }}{\text{Hypotenuse }}$Opposite Hypotenuse ?
Question 2
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.
Question 3
Find the value of $f$f, correct to two decimal places.
Question 4
Find the value of $x$x to the nearest degree.
A right-angled triangle with vertices labeled A, B and C. Vertex A is at the top, B at the bottom right, and C at the bottom left. A small square at vertex A indicates that it is a right angle. Side interval(BC), which is the side opposite vertex A, is the hypotenuse and is marked with a length of 25. The angle located at vertex B is labelled x. Side interval(AB), descending from the right angle at vertex A to vertex B, is marked with a length of 7, and is adjacent to the angle x. Side interval(AC) is opposite the angle x.