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Find cos 𝐴, given 𝐴𝐵𝐶 is a right-angled triangle at 𝐶 where 𝐴𝐵 = 10 cm and 𝐵𝐶 = 6 cm.
Find cos of 𝐴, given 𝐴𝐵𝐶 is a
right-angled triangle at 𝐶, where 𝐴𝐵 is equal to 10 centimeters and 𝐵𝐶 is equal
to six centimeters.
This question is asking us to find
the cos or cosine of an angle 𝐴. The definition of the cosine or cos
of an angle 𝜃 in a right-angled triangle is the ratio between the adjacent side and
the hypotenuse of the triangle. Let’s recall what these terms
The hypotenuse in any right-angled
triangle is the side directly opposite the right angle. It’s also the longest side of the
triangle. The side opposite the angle we’re
interested in — so in this case that’s angle 𝐴 — is called the opposite. The third side of the triangle
which is between the angle we’re interested in and the right angle is called the
adjacent. This means that in our triangle cos
of the angle 𝐴 is equal to 𝐴𝐶, the adjacent side, divided by 𝐴𝐵, the
We’ve been given the length of the
hypotenuse is 10 centimeters. So we have that cos of angle 𝐴 is
equal to 𝐴𝐶 over 10. In order to find cos of angle 𝐴,
we need to work out the length of 𝐴𝐶. Notice that we’ve 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
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
𝐴𝐶 squared plus 𝐵𝐶 squared is equal to 𝐴𝐵 squared. 𝐵𝐶 is six centimeters and 𝐴𝐵 is
10 centimeters. So we have the equation 𝐴𝐶
squared plus six squared is equal to 10 squared. And we can solve this equation to
find the length of 𝐴𝐶.
Six squared is equal to 36 and 10
squared is equal to 100. So we have 𝐴𝐶 squared plus 36 is
equal to 100. Subtracting 36 from each side of
the equation, we have that 𝐴𝐶 squared is equal to 64. To find the value of 𝐴𝐶, we need
to take the square root. 64 is a squared number. So its square root is just
Now, you may have been able to spot
this without going through all the working out for the Pythagorean theorem if you’re
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
𝐴𝐶 the adjacent side is eight centimeters. So can substitute this into our
formula for cos of 𝐴. We have that cos of 𝐴 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 𝐴 is equal to four-fifths.
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