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Study Advice Service Student Support Services ‘Mini’ Version: a more detailed version is available on our main maths resources page. Trigonometry and Pythagoras’ Theorem Pythagoras’ Theorem Pythagoras’ Theorem is used to find the lengths of unknown sides in right-angled triangles. The rule is: a 2 + b 2 = c 2 In words: The square of the hypotenuse is equal to the sum of the squares of the other two sides. c b Here a and b are the two shorter sides of the triangle - the ones which are attached to the right-angle. c or h is the hypotenuse, the longest side; the side that lies opposite the right-angle. a Trigonometry Basic trigonometry uses the rules sine, cosine and tangent. The most common use of sine, cosine and tangent is with right-angled triangles. They are used to find unknown sides and angles. These functions are reliant on either knowing an angle and a side or the lengths of two sides. The formulae for sine, cosine and tangent are: sin θ = opposite hypotenuse cos θ = adjacent hypotenuse tan θ = opposite adjacent Where θ is used to denote the angle of interest, and sinθ , for example, is the value of the sine function acting on θ . Given the right-angled triangle below, how can the length of side c be found? An angle and a side are known. Look for a trigonometric formula which includes both the known side and the unknown side. c opposite 5 sinθ = includes all of the necessary information as the side hypotenuse opposite the angle is known and the hypotenuse is the side that is to 30° be determined. Substituting the values in gives: sin 30° = 5 . Rearrange: c × sin 30° = 5 ⇒ c = 5 c sin 30° All that remains is to substitute in the value of sin 30° (found via your calculator) and work out the value of the fraction. c = 5 = 5 = 10 . Hence the length of side c is 10. sin 30° 0.5 Given the right-angled triangle below, how do we find the size of angle θ ? 12 5 θ We can use the same formulae as we have been using for finding unknown sides. In the above triangle the sides that are known are, in relation to θ , the opposite side and the hypotenuse. So, use a formula which uses both the opposite side and the hypotenuse. sinθ = opposite can hypotenuse be used here. Substituting in the values of the known sides gives: sin θ = 5 . 12 sin−1( 5 ) = 24.62° to 2 d.p. So θ = 24.62° (Note sin−1 can be found on your calculator via the 12 shift or 2nd key.) Tel: 01482 466199 Web: www.hull.ac.uk/studyadvice Email: [email protected]