Video Transcript
Find cos of ๐ด, given ๐ด๐ต๐ถ is a
right-angled triangle at ๐ถ, where ๐ด๐ต is equal to 10 centimeters and ๐ต๐ถ is equal
to six centimeters.
This question is asking us to find
the cos or cosine of an angle ๐ด. The definition of the cosine or cos
of an angle ๐ in a right-angled triangle is the ratio between the adjacent side and
the hypotenuse of the triangle. Letโs recall what these terms
mean.
The hypotenuse in any right-angled
triangle is the side directly opposite the right angle. Itโs also the longest side of the
triangle. The side opposite the angle weโre
interested in โ so in this case thatโs angle ๐ด โ is called the opposite. The third side of the triangle
which is between the angle weโre interested in and the right angle is called the
adjacent. This means that in our triangle cos
of the angle ๐ด is equal to ๐ด๐ถ, the adjacent side, divided by ๐ด๐ต, the
hypotenuse.
Weโve been given the length of the
hypotenuse is 10 centimeters. So we have that cos of angle ๐ด is
equal to ๐ด๐ถ over 10. In order to find cos of angle ๐ด,
we need to work out the length of ๐ด๐ถ. Notice that weโve been given the
length of both of the other two sides of the triangle. And as the triangle is right
angled, we can apply the Pythagorean theorem to calculate the length of the third
side.
The Pythagorean theorem tells us
that in a right-angled triangle, the sum of the squares of the two shorter sides is
equal to the square of the hypotenuse. In this triangle, this means that
๐ด๐ถ squared plus ๐ต๐ถ squared is equal to ๐ด๐ต squared. ๐ต๐ถ is six centimeters and ๐ด๐ต is
10 centimeters. So we have the equation ๐ด๐ถ
squared plus six squared is equal to 10 squared. And we can solve this equation to
find the length of ๐ด๐ถ.
Six squared is equal to 36 and 10
squared is equal to 100. So we have ๐ด๐ถ squared plus 36 is
equal to 100. Subtracting 36 from each side of
the equation, we have that ๐ด๐ถ squared is equal to 64. To find the value of ๐ด๐ถ, we need
to take the square root. 64 is a squared number. So its square root is just
eight.
Now, you may have been able to spot
this without going through all the working out for the Pythagorean theorem if youโre
familiar with your Pythagorean triples as six, eight, 10 is an example of one. A Pythagorean triple is a
right-angled triangle in which the length of all three sides are integers. In fact, the six, eight, 10
triangle is just an enlargement of the three, four, five triangle, which is probably
the most commonly recognized Pythagorean triple.
Whether you spot it straight away
or whether you had to go through the working out, we now know that the length of
๐ด๐ถ the adjacent side is eight centimeters. So can substitute this into our
formula for cos of ๐ด. We have that cos of ๐ด is equal to
eight over 10. This fraction can be simplified
slightly as both the numerator and denominator are multiples of two. So dividing them both by two, we
have that cos of ๐ด is equal to four-fifths.
