Teach Secondary - Issue 13.2

[ M A T H S P R O B L E M ] Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University. He has written many books and articles for mathematics teachers. foster77.co.uk , blog.foster77.co.uk THE DIFFICULTY Shaun and Emily want to work out the length of the horizontal side in this triangle. In this lesson, students explore the relationships between the acute angles and the relevant sides in right-angled triangles What kind of triangle is it? Right-angled. Which side is the horizontal side? The one at the bottom. Shaun says he needs to know the size of the red angle to work it out. Emily says she needs to know the size of the blue angle to work it out.Who is right? In fact, either angle can be used, depending on whether they use cosine (red angle) or sine (blue angle). THE SOLUTION What are the coloured labels on this diagram showing? These labels are showing that: • the horizontal side is adjacent to the red angle and opposite to the blue angle. • the vertical side is adjacent to the blue angle and opposite to the red angle. Write adj and adj and opp and opp on the sides of these triangles: Estimate possible values for the sizes of the red and blue angles in Shaun and Emily’s triangles. Go with whatever sensible sizes are suggested (e.g. red = 30°, blue = 60°). Here is what Shaun wrote, using the red angle: What did Emily write, using the blue angle? Emily wrote 8 sin 60° on the horizontal side and 8 cos 60° on the vertical side. This helps students to see that, in general, sin(90° − ) = cos and cos(90° − ) = sin , so there are always two ways to find each side. Do the same thing for the three other triangles given above. Checking for understanding A right-angled triangle has a hypotenuse of 10 cm and one of its angles 35°. Find the lengths of both of the other sides in two different ways for each . The answers are 5.74 cm and 8.19 cm (correct to 2 decimal places). TRIGONOMETRY IN RIGHT-ANGLED TRIANGLES Students often muddle up the different sides when doing trigonometry in right-angled triangles, says Colin Foster 8 cm 8 cm adj = opp opp = adj 8 cm 8 cos 30° 8 sin 30° 6 cm 12 cm 5 cm 21 teachwire.net/secondary