When considering right-angled triangles we need to know which side is the hypotenuse (the side opposite the right angle). Then the sides which we label as opposite and adjacent sides are always in relation to a particular angle.
Finding side lengths
If we know two sides and want to find the third, we can use Pythagoras' theorem.
$a^2+b^2=c^2$a2+b2=c2
If we know one side length and an angle, we can use one of the 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 SOH-CAH-TOA 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 our knowledge of 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 $f$f, correct to two decimal places.
A right-angled triangle with a 29-degree opposite to its base is shown. The side adjacent the 29-degree angle is labeled as "$f$f cm" indicating a length in centimeters that is not specified. The hypotenuse is labeled "7 cm," indicating a length of 7 centimeters.
The side opposite to the 29-degree angle is not labeled. The right angle is indicated by a small square at the triangle's base
Question 3
Find the value of $h$h, correct to two decimal places.
Finding the size of an unknown angle
We can also use the trigonometric ratios to find the size of unknown angles. To do this we need any $2$2of the side lengths.
We identify which ratio we need to use, we write the rule, we fill in the information which we know then use inverse operations to rearrange and solve.
Practice questions
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.
Problem solving with right-angled triangles
We can use trigonometry in real world applications or in solving geometrical problems. We also come across the angle of elevation and depression in applications of right-angled triangles.
| Angle of Elevation: the angle made between the line of sight of the observer and the 'horizontal' when the object is ABOVE the horizontal (observer is looking UP) |  |
| Angle of Depression: the angle made between the line of sight of the observer and the 'horizontal' when the object is BELOW the horizontal (observer is looking DOWN) |  |
Practice questions
Question 5
A ladder measuring $2.36$2.36 m in length is leaning against a wall. If the angle the ladder makes with the ground is $y$y$^\circ$°, find $y$y to two decimal places.
Question 6
Consider the given figure.
Find the unknown angle $x$x, correct to two decimal places.
Find $y$y, correct to two decimal places.
Find $z$z correct to two decimal places.
Question 7
Jack is standing at the tip of a tree's shadow and knows that the angle from the ground to the top of the tree is $34^\circ$34°.
If Jack is standing $29$29 metres away from the base of the tree, what is the value of $h$h, the height of the tree to the nearest two decimal places?
question 8
In the following diagram, $\angle CAE=61^\circ$∠CAE=61°, $\angle CBE=73^\circ$∠CBE=73° and $CE=25$CE=25.
Find the length of $AE$AE, correct to four decimal places.
Find the length of $BE$BE, correct to four decimal places.
Hence, find the length of $AB$AB, correct to two decimal places.
Find the length of $BD$BD, correct to one decimal place.