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.
$\sin\theta$sinθ = $\frac{Opposite}{Hypotenuse}$OppositeHypotenuse = $\frac{O}{H}$OH
$\cos\theta$cosθ = $\frac{Adjacent}{Hypotenuse}$AdjacentHypotenuse = $\frac{A}{H}$AH
$\tan\theta$tanθ = $\frac{Opposite}{Adjacent}$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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From similar triangles we know that two triangles with the same angles will be similar and the ratio of corresponding sides will be equal. 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.
Practice questions
Question 1
Find the value of $f$f, correct to two decimal places.
Question 2
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.
Question 3
Find the value of $x$x, the side length of the parallelogram, to the nearest centimetre.
A parallelogram with its angle at upper-right corner labeled as 52 degrees, indicating its measure. Other angles of the parallelogram are not labeled. Its top side is labeled 32 cm, indicating its length. Its left side is labeled $x$x cm, indicating its unknown length. A perpendicular internal segment is drawn from the upper-left corner to the bottom side of the parallelogram, thus creating a right-angled triangle on the left side. The bottom side of the parallelogram, which is parallel to the top side as indicated by the double arrowheads, is cut into two parts: the left part with no measurement indicated and the right part which measures 11 cm, as labeled.
In the triangle, the sides are the internal segment, the left side of the parallelogram, and the left part of the bottom side of the parallelogram. A square symbol is shown at the corner where the internal segment and the bottom side intersect to indicate that it is a right angle. The left side of the parallelogram labeled $x$x cm acts as the hypotenuse of the triangle. The internal segment is the vertical leg of the triangle. The left part of the bottom side of the parallelogram acts as horizontal leg of the triangle.
Special triangles
We can create right-angled triangles of varying side lengths and angle combinations. There are, however, 2 very special triangles that are referred to often in trigonometric studies. These triangles are called exact value triangles, and they look like this.
Observations
- Any right-angled triangle with a $45^\circ$45° angle will be isosceles. This means it will have two equal sides (here we have the simplest case where they measure $1$1 unit each). How can we obtain the hypotenuse?
- The right-angled triangle with $30^\circ$30° and $60^\circ$60° angles can be obtained by cutting an equilateral triangle in half. See if you can start with an equilateral triangle of side length $2$2 to obtain the exact side lengths in the above triangle.
From these 2 triangles, we can construct trigonometric ratios for the angles$30$30, $45$45 and $60$60 degrees.
Now, an isosceles right-angled triangle may not have its sides measuring $1$1,$1$1 and $\sqrt{2}$√2, but however large it is, it will always have two $45^\circ$45° angles and the ratios of the sides will always be the same as in the table. The same applies to the triangle with $60^\circ$60° and $30^\circ$30° angles.
These particular values are ones that we need to learn for our continued study in high school trigonometry, as they will help us obtain exact rather than rounded values. It may be easier to learn the two triangles, than to try to learn the tables of values.
Practice questions
Question 4
Find the value of $\theta$θ if $\sin\theta=\frac{\sqrt{3}}{2}$sinθ=√32, given that $0^\circ\le\theta\le90^\circ$0°≤θ≤90°.
Question 5
Given that $\tan\theta=\frac{1}{\sqrt{3}}$tanθ=1√3, find $\sin\theta$sinθ.
First, find the value of $\theta$θ, given that $0^\circ\le\theta\le90^\circ$0°≤θ≤90°.
Hence, find the value of $\sin\theta$sinθ to two decimal places.
Problem solving with right-angled triangles
When right angles are involved don't forget we can always use Pythagoras' theorem to find an unknown side given the other two sides. And in worded problems in trigonometry, the angle of elevation or depression is often used to describe the situation.
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Pythagoras' Theorem: $a^2+b^2=c^2$a2+b2=c2, where c is the hypotenuse
Angle of Elevation: the angle from the observer's horizontal line of sight looking UP at an object
Angle of Depression: the angle from the observer's horizontal line of sight looking DOWN at an object
Practice questions
Question 6
A man standing at point $C$C, is looking at the top of a tree at point $A$A. Identify the angle of elevation in the figure given.
A right triangle is shown with vertices labeled A, B and C. Side AB is the vertical leg, side BC is the horizontal leg, and side AC is the hypotenuse. The right angle is at vertex B, as indicated by a small square. There are two arcs indicating angles: one at vertex C, labeled with $\alpha$α, and another at vertex A, labeled with $\theta$θ. Above vertex C, a vertical dotted line extends from vertex C to a point labeled D. The angle between this dotted line CD and the hypotenuse AC is labeled with $\sigma$σ.
$\alpha$α
A$\theta$θ
B$\sigma$σ
C
Question 7
From the top of a rocky ledge $188$188 m high, the angle of depression to a boat is $13^\circ$13°. If the boat is $d$d m from the foot of the cliff find $d$d correct to two decimal places.
Question 8
Consider the following diagram.
Find $y$y, correct to two decimal places.
Find $w$w, correct to two decimal places.
Hence, find $x$x, correct to one decimal place.