Similarity, Right Triangles, and Trigonometry
Introduction to trigonometry
Find sine, cosine and tangent in a right-angled triangle and recall the standard angles.
Take a right-angled triangle and pick one of the two non-right angles. The side across from it is the opposite, the side beside it is the adjacent, and the long side facing the right angle is the hypotenuse. Trigonometry is about the three ratios you can build from those sides.
Have a play
One angle of this right-angled triangle has been marked. Find out what that angle and each of the three sides are called.
Tap any part of the picture.
SOH CAH TOA
Sine is Opposite over Hypotenuse. Cosine is Adjacent over Hypotenuse. Tangent is Opposite over Adjacent. Say it once out loud and you have the whole topic in nine letters.
The surprise is that these ratios depend only on the angle, never on how big the triangle is. Any two right-angled triangles with a 30° angle are similar, so their sides are in the same proportion — which is why sin 30° is always exactly 1/2, whether the triangle fits on your thumbnail or spans a field.
Worked example
A right-angled triangle has sides 3, 4 and 5. Find the three ratios for the angle opposite the side of 3.
Label the sides: opposite = 3, adjacent = 4, hypotenuse = 5.
The hypotenuse is always the longest side, and the opposite is the one that does not touch the angle at all.
Try it together
A right-angled triangle has an opposite side of 5 and a hypotenuse of 13. Let us find the cosine of that angle.
Cosine needs the adjacent side, and we have not been given it. Pythagoras will hand it over.
1.Pythagoras says adjacent² = hypotenuse² - opposite². What is 169 - 25?
Have a go
Have a go on your own: in that same triangle, what is the tangent of the angle? Give your answer as a fraction.
Hint: Opposite over adjacent, so 5 over 12.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.