Video: Finding the Cosine of Angles in Right-Angled Triangles given the Opposite Side and the Hypotenuse

Video Transcript

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 mean.

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 hypotenuse.

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 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 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 eight.

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.