Trigonometry & Pythagoras Theorem

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1 Trigonometry & Pythagoras Theorem Mathematis Skills Guide This is one of a series of guides designed to help you inrease your onfidene in handling Mathematis. This guide ontains oth theory and exerises whih over:-. Pythagoras Theorem. Introdution to trigonometry. Using trigonometry to find an unknown side 4. Using trigonometry to find an unknown angle 5. Trigonometri diagrams and identities There are often different ways of doing things in Mathematis and the methods suggested in the guides may not e the ones you were taught. If you are suessful and happy with the methods you use it may not e neessary for you to hange them. If you have prolems or need help in any part of the work then there are a numer of ways you an get help. For students at the University of Hull Ask your leturers. You an ontat a maths Skills Adviser from the Skills Team on the shown elow. Aess more maths Skills Guides and resoures at the wesite elow. Look at one of the many textooks in the lirary. We: skills@hull.a.uk

2 . Pythagoras Theorem Pythagoras Theorem is used to find the lengths of unknown sides in triangles. It an e used in the following onditions:. The triangle is a right-angled triangle, i.e. ontains an angle of 90 degrees.. Two of the three sides are already known. Two main uses of Pythagoras Theorem are onverting a vetor to magnitude and diretion form and finding the resultant fore given a fore in the horizontal and vertial diretions. The rule is: In words: a The square of the hypotenuse is equal to the sum of the squares of the other two sides. Note that this is sometimes expressed as a Examples Finding the hypotenuse a h. Here a and are the two shorter sides of the triangle - the ones whih are attahed to the right-angle. or h is the hypotenuse, the longest side; the side that lies opposite the right-angle. 7 If a, 7 find. Sustitute the known values into Pythagoras Theorem: 7 Evaluate the Left Hand Side (LHS) Take the positive square root of oth sides: 58 This an now e left as 58 or a alulator may e used to find to d.p. Finding a shorter side If a 5, find. 5 Sustitute the known values into Pythagoras Theorem: 5 This eomes Sutrat 5 from oth sides to get on its own: Take the positive square root of oth sides: 44. So.

3 Pythagoras Theorem is often used to find the length of vetors. The theorem an e extended to dimensions y squaring all omponents and adding, then square rooting. For more information on this, please refer to Vetors, availale from Exerise For eah of the following triangles find the length of the hypotenuse: a) ) ) For eah of the following triangles find the length of the unknown side: a) ) ) 4 0 a 6 7. Introdution to trigonometry Basi trigonometry uses the rules sine, osine and tangent. These funtions are atually infinite series and would prove very diffiult and time-onsuming to alulate to a reasonale degree of auray y hand. Fortunately sientifi alulators are ale to deal with these funtions. The most ommon use of sine, osine and tangent is with right-angled triangles. They are used to find unknown sides and angles. These funtions are reliant on either knowing an angle and a side or the lengths of two sides. The formulae for sine, osine and tangent are: opposite adjaent opposite sin os tan hypotenuse hypotenuse adjaent Where is used to denote the angle of interest, and sine funtion ating on. These rules are often rememered y writing down SOH CAH TOA sin, for example, is the value of the You may find it useful to inlude a / etween the seond and third letters in eah row to remind you of the division. Hypotenuse, adjaent and opposite refer to the lengths of the sides of the triangle. Note that whilst the position of the hypotenuse is fixed (it is always the side opposite the right angle), the positions of the opposite and adjaent sides are dependant on the loation of the angle that is eing used.

4 Hypotenuse Opposite Hypotenuse Angle Adjaent Angle Adjaent Opposite The position of the angle of interest determines the laels on the sides.. Using trigonometry to find an unknown side Trigonometry an e used to find an unknown side of a triangle when you know only one angle and the length of one side. Given a right-angled triangle suh as: 5 0 How an the length of side e found? An angle and a side are known. Look for a trigonometri formula whih inludes oth the known side and the unknown side. opposite sin inludes all of the neessary information as the side opposite the angle hypotenuse is known and the hypotenuse is the side that is to e determined. Sustituting the values in gives: sin0 5. Rearrange: sin0 5 5 sin0 (for help with rearranging equations see Algera ) All that remains is to sustitute in the value of sin0 (found via your alulator) and work out the value of the fration sin Hene the length of side is 0. Another example: 45

5 Here we use osine, as the side we need is adjaent to the known angle, and the hypotenuse is known. So os 45, rearranging, os Notes: Rememer to hek that your alulator is in degrees if using degrees or in radians if using radians. This an normally e altered via the mode utton. Depending on the make and age of the alulator eing used it may e neessary to type in either sin 0 or 0 sin to get the value of sin 0. Always work with the numers as they are shown on the alulator sreen until the final result is produed. Then this figure an e rounded. Rounding figures part-way through the alulation will result in a less aurate answer. Exerise Find the lengths of the missing sides in the following triangles: a) ) ) 60 5 d e 75 0 a 8 f 4. Using trigonometry to find an unknown angle Trigonometry an e used to find an unknown angle of a right-angled triangle when you know only the length of two sides. Given a right-angled triangle suh as: 5 How do we find the size of angle? We an use the same formulae as we have een using for finding unknown sides. In the aove triangle the sides that are known are, in relation to, the opposite side and the hypotenuse. So, use a formula whih uses oth the opposite side and the hypotenuse. opposite sin an e used here. hypotenuse Sustituting in the values of the known sides gives: sin To get from sin 0. 4 to a value for you need the inverse sine operation.

6 This is on most alulators as sin and is usually aessed using a nd funtion or shift key, then the sin key. sin ( 5 ) 4. 6 to d.p. So 4. 6 Note that I used 5 in the alulation rather than 0.4. This is eause 0.4 is rounded and so is less aurate than 5. Another example: use 6 6 Here the known sides are the adjaent and osine. os So, os ( 0. 75) to d.p. the hypotenuse, so we Notes: Rememer to hek that your alulator is in degrees if you are using degrees or in radians if you are using radians. This an normally e altered via the mode utton. Depending on your alulator you may need to type in either nd /shift sin 0 or 0 nd /shift sin to get the value of sin 0. Exerise Find the size of the marked angles a) ) ) Trigonometri diagrams and identities Diagrams There are diagrams that an e memorised in order to reall ertain values of sin, os and tan quikly. The first is a right-angled isoseles triangle with two sides of length. Beause this is an Isoseles triangle, it has angles the same. The size of these angles is

7 Hene this triangle will provide us with the values for os 45, sin 45, and tan 45. Using Pythagoras Theorem, we find that the length of the hypotenuse is equal to Using the formulae for sin, os and tan on either angle, we an now find that: os 45 sin45 tan45 You may wish to onfirm these answers yourself.. The seond diagram is used to find sin, os and tan for angles of 0 and 60 It is half of an equilateral triangle with sides of length This gives us angles of 0 and 60 as the angles of an equilateral triangle are all 60. Here we know the length of the hypotenuse, ut one of the shorter sides is unknown. Using Pythagoras Theorem we get 4, hene the length of the missing side is. Using the formulae for sin, os and tan, we an now find that: os0 sin0 tan0 and os60 sin60 tan60 Again, you may wish to hek these. Notes: If you have angles that add up to 90, then the sine of the first will equal the osine of the seond, for example see sin60 and os0. It is est to leave in the surds or square-root signs. Results suh as answers, rounding them off will only make them less aurate. are exat Identities There are a numer of identities in trigonometry. Identities are fats that will always e true, no matter whether numers (or in this ase angles) are hanged. These an e used to simplify long algerai arguments. The one you are most likely to enounter is: sin os

8 This an e seen using a diagram: If you are given a right-angled triangle with hypotenuse length, then the adjaent side an e written as hyp os y rearranging the formula for osine and the opposite side as hyp sin y rearranging the sine rule. Sine the hypotenuse in this ase is, these simplify to os and sin. Using Pythagoras Theorem, the squares of these sides must sum to give the square of the hypotenuse, hene sin os Other identities whih may e of use are: sin( A B) sin Aos B os Asin B os( A B) os Aos B sin Asin B (Note the sign) os A os A sin A sina sin Aos A Answers Exerise. a) 0 ).89 to d.p. ) 9.85 to d.p.. a). to d.p. ) 9.08 to d.p. ) 0.95 to d.p. Exerise. a 9.5 to d.p., to d.p., d 5.60 to d.p. e 0.5 to d.p., f.68 to.d.p. Exerise a) ) 48.9 ) (all to d.p.) We would appreiate your omments on this worksheet, espeially if you ve found any errors, so that we an improve it for future use. Please ontat the Maths Skills Adviser y at skills@hull.a.uk The information in this leaflet an e made availale in an alternative format on request using the address aove.