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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.)
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