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GY1004: EARTH A DYNAMIC PLANET B BASIC NUMERACY: AREFRESHER

HELP SHEET 1 BASIC TRIGONOMETRY

1.1 Trigono!"r#.1.1.1 Introduction. Geographers often use calculations involving right-angletriangles in, for example, calculating the height of a tree or cli using aclinometer. The study of triangles is called trigonometry.

There are two types of problems involving right-angle triangles that you willcome across:

1 those where you !now the length of two sides and "ust want to #nd the lengthof the remaining side without needing the angles $ for this you can use%ythagoras& theorem'

( those where you either !now angles and need to #nd side lengths, )* you!now side lengths and need to #nd angles $ for this you need to use thetrigonometric ratios, sine, cosine or tangent.

+.. or calculations involving triangles that do not have a right-angle you willneed to use dierent euations that are a bit more complicated: we won&tconsider these.

1.1.2 Internal angles of a triangle. /n important rule for all triangles worthremembering is that "$! in"!rn%& %ng&!' o( % "ri%ng&! %&)%#' %** +, "o1-0.0ince a right-angle is 23, this means that the other two angles inside aright angle triangle must add up to 23 as well. Therefore if you !now one of the

angles 4let&s call it ' the Gree! letter 5alpha& you can uic!ly calculate the other4let&s call it ' the Gree! letter 5beta&, because 6 7 2 and rearranging thisgives us 7 2- . 8f you are not familiar with the commonly used symbols fromthe Gree! alphabet, see 9elp 0heet (.

1.1.3 The hypotenuse. The longest side, !nown as the $#,o"!n+'!, is alwaysthe other side of the triangle from the right angle. ou can thin! of it as the rightangle symbol always pointing across the triangle to the hypotenuse.

right angle

hypotenuse

1./ C%&+&%"ing "$! &!ng"$ o( % 'i*! +'ing P#"$%gor%' "$!or!. 8f thelength of two sides of a right angled triangle are !nown, the length of the thirdside can be calculated using %ythagoras& theorem even if you don&t !now any ofthe angles 4though it must be a right-angled triangle.

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%ythagoras states that the suare of the hypotenuse 49( is eual to the sum ofthe suares of the other two sides $ the opposite and ad"acent 4)(6 /(, or in aneuation:

9(7 )(6/(

8f you !now two of the sides, then you can substitute the !nown values into theeuation and solve to #nd the un!nown.

8n any trigonometric calculations, it is useful to study the problem and then drawa uic! s!etch of the triangle, drawing angles that you !now approximatelycorrectly and side lengths approximately to scale. /nnotate all the informationthat you !now already onto your s!etch, ma!ing sure to mar! the right angle andthe hypotenuse.

or example:

;(7 ((6

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tan 4angle 7 oppositeBad"acent tan 4angle 7 )B/

and are only applicable to right angle triangles. These ratios can beremembered by representing them as strings of letters, as in 0)9-=/9-T)/. ThememoriCation of this mnemonic can be aided by expanding it into a phrase, suchas 50ome )Dcers 9ave =urly /uburn 9air Till )ld /ge& 4or replace with your ownphrase constructed of words beginning with the letters 0-)-9-=-/-9-T-)-/E

=learly, in using these rules it is important to identify the opposite and ad"acentsides of the triangle. This is where it is easy to get confused because which sideis opposite and which side is ad"acent depends on which angle you areexaminingE Tip pretend you are standing inside the angle in question andlooking out from it. The opposite face will be across from you, the ad"acent sidewill be to your side.

hypotenuse

opposite

adjacent

adjacent

hypotenuseopposite

Fecide which of the trigonometric ratios to use $ you need to use the one thathas two !nown values along with the un!nown value that you are trying to #nd.

=onsider:

? m3 m

hypotenuse45

9ere, the !nown length 4A m is ad"acent to the angle ?;3, and we want to #ndthe length of the hypotenuse, therefore we use cosine

0ubstitute your values into the cosine euation and rearrange to #nd theun!nown:

cos ?;3 7 AB9

ultiply both sides by 9 Hcos45 =3 H

H

=ancel 9 which is top and bottom of *90 Hcos45 = 3

Fivide by cos ?;3 4note moves as a single term Hcos 45

cos45=

3

cos 45

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=ancel cos ?;3 which is top and bottom on H90 H=3

cos45

>valuate your euation.8n our example that means using your calculator to #ndthe cosine of ?;3 and dividing A by that value 7 ?.(?A m. IMPORTANT: efore

your start doing anything with sine, cosine or tangent chec! that your calculatoris using degrees to calculate angles - your calculator should show 5F>G&somewhere on the screen. The alternative is another measure of angle calledradians that will cause problems if your calculator thin!s you are giving it anglesin radians rather than degrees.

+ote that in the situation where you have two lengths and are trying to #nd theangle you will not need to rearrange the euation, but you need to use theinverse sin 47sin-1, cos 47cos-1 or tan 47tan-1 button on your calculator to #ndthe angle from the ratio of the two sides. or example:

cos I 7 /B9

cos I 7 AB?.(?A 7 2.J2J1I 7 cos-12.J2J1 7 ?;3

4.243 m3 m

hypotenuse?

The inverse trigonometric functions 4cos-1, sin-1 and tan-1 produce the anglecorresponding to a particular value for a cosine, sine or tangent. or example,sin4AJ K 7 2.L2( and therefore sin-142.L2( 7 AJ K . e careful: althoughstandard, this notation is poor since the suare of a sin 4or tan or cos is writtensin( 4e.g. the suare of sin4?; is sin(4?;. Thus sin-14?; might be thought

erroneously to be 1Bsin4?;. 8t is not.