Right-angled triangles A right-angled triangle contains a right angle. The longest side opposite the right angle is called the hypotenuse. Teacher notes Review the definition of a right-angled triangle. Explain that the hypotenuse is the side that does not touch the right angle. Tell pupils that in any triangle the angle opposite the longest side will always be the largest angle and vice-versa. Ask pupils to explain why no other angle in a right-angled triangle can be larger than or equal to the right angle. Ask pupils to tell you the sum of the two smaller angles in a right-angled triangle. Recall that the sum of the angles in a triangle is always equal to 180°. Recall, also, that two angles that add up to 90° are called complementary angles. Conclude that the two smaller angles in a right-angled triangle are complementary angles.

The opposite and adjacent sides The two shorter sides of a right-angled triangle are named with respect to one of the acute angles. The side opposite the marked angle is called the opposite side. x The side between the marked angle and the right angle is called the adjacent side. Teacher notes You don’t need to know the actual size of the marked angle to label the two shorter sides as shown. It could be labelled using a letter symbol. Here we use x. Point out that if we labelled the sides with respect to the other acute angle their names would be reversed.

Label the sides Teacher notes Use this activity to practice labelling the sides of a right-angled triangle with respect to angle θ. Ask volunteers to come to the board to complete the activity.

Similar right-angled triangles If two right-angled triangles have an acute angle of the same size they must be similar. For example, two triangles with an acute angle of 37° are similar. 8 cm 6 cm 10 cm 37° 3 cm 4 cm 5 cm 37° Teacher notes Point out that if one acute angle is the same size in both triangles, then all three angles must be the same size in both triangles, because one angle is a right angle and the other angle is therefore the complement of the given acute angle. Remind pupils that a ratio compares one part a to another part b. We can write a ratio as a:b or a/b. The ratio of the side lengths in each triangle is the same. 3 4 = 6 8 opp adj 3 5 = 6 10 opp hyp 4 5 = 8 10 adj hyp

Similar right-angled triangles Teacher notes Drag point B to show that when the triangle is made larger or smaller, but angle A stays the same, then the ratios do not change. Click the blue box to change the formula around.

Trigonometry The word trigonometry comes from the Greek meaning ‘triangle measurement’. Trigonometry uses the fact that the side lengths of similar triangles are always in the same ratio to find unknown sides and angles. For example, when one of the angles in a right-angled triangle is 30° the side opposite this angle is always half the length of the hypotenuse. 6 cm 12 cm ? Teacher notes Stress that no matter how big a right angled triangle is, if one of the angles is 30° the side opposite the 30° angle will always be half the length of the hypotenuse. The converse is also true. If a right-angled triangle has one side that is half the length of the hypotenuse then the angle opposite that side must be 30°. 30° 8 cm ? 4 cm 30°

The sine ratio the length of the opposite side The ratio of is the sine ratio. the length of the hypotenuse The value of the sine ratio depends on the size of the angles in the triangle. O P S I T E H Y P O T E N U S opposite Teacher notes The sine ratio depends on the size of the opposite angle. We say that the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse. Sin is mathematical shorthand for sine. It is still pronounced as ‘sine’. sin θ = hypotenuse θ

The sine ratio What is the value of sin 65°? 65° This is the same as asking: In a right-angled triangle with an angle of 65°, what is the ratio of the opposite side to the hypotenuse? To work this out we can accurately draw a right-angled triangle with a 65° angle and measure the lengths of the opposite side and the hypotenuse.

The sine ratio What is the value of sin 65°? It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 65° are similar. The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same and we have measured accurately. In this triangle, 65° 10 cm 11 cm opposite Teacher notes This ratio can also be demonstrated using the similar right-angled activity on slide 7. sin 65° = hypotenuse 10 = 11 = 0.91 (to 2 d.p.)

The sine ratio using a table What is the value of sin 65°? It is not practical to draw a diagram each time. Before the widespread use of scientific calculators people would use a table of values to work this out. Here is an extract from a table of sine values: Angle in degrees 0.0 0.1 0.2 0.3 0.4 0.5 Teacher notes Discuss how to find the value of sin 65° using the table. Discuss how other values can be found from the table. The table appears to show that the sine of 64.1° is the same as the sine of 64.2°. Explain that these appear to be the same because the values in the table are rounded to 3 decimal places. The sine of 64.1° is actually slightly less than the sine of 64.2°. We can find the values to a higher degree of accuracy using a scientific calculator. If possible, allow pupils to look at some actual tables of trigonometric functions. Notice that for the sine ratio, the bigger the angle the bigger the ratio of the opposite side over the hypotenuse. Ask pupils to explain why the sine of an angle between 0° and 90° will always be between 0 and 1. 63 0.891 0.892 0.893 0.893 0.894 0.895 64 0.899 0.900 0.900 0.901 0.902 0.903 65 0.906 0.907 0.908 0.909 0.909 0.910 66 0.914 0.914 0.915 0.916 0.916 0.917

The sine ratio using a calculator What is the value of sin 65°? To find the value of sin 65° using a scientific calculator, start by making sure that your calculator is set to work in degrees. Key in: sin 6 5 = The calculator should display 0.906307787 This is 0.906 to 3 significant figures. Some calculators require you to put in the angle first: 65 sin. In this case you do not need the equals sign.

The cosine ratio the length of the adjacent side The ratio of is the cosine ratio. the length of the hypotenuse The value of the cosine ratio depends on the size of the angles in the triangle. H Y P O T E N U S adjacent cos θ = hypotenuse θ A D J A C E N T

The cosine ratio What is the value of cos 53°? 53° This is the same as asking: In a right-angled triangle with an angle of 53°, what is the ratio of the adjacent side to the hypotenuse? To work this out we can accurately draw a right-angled triangle with a 53° angle and measure the lengths of the adjacent side and the hypotenuse.

The cosine ratio What is the value of cos 53°? It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 53° are similar. The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same. In this triangle, adjacent Teacher notes This ratio can also be demonstrated using the similar right-angled activity on slide 7. cos 53° = hypotenuse 10 cm 6 = 10 53° = 0.6 6 cm

The cosine ratio using a table What is the value of cos 53°? Here is an extract from a table of cosine values: Angle in degrees 0.0 0.1 0.2 0.3 0.4 0.5 50 0.643 0.641 0.640 0.639 0.637 0.636 51 0.629 0.628 0.627 0.625 0.624 0.623 52 0.616 0.614 0.613 0.612 0.610 0.609 53 0.602 0.600 0.599 0.598 0.596 0.595 Teacher notes Discuss how to find the value of cos 53° using the table. Discuss how other values can be found from the table. If possible, allow pupils to look at some actually tables of trigonometric functions. Notice that for the cosine ratio, the bigger the angle the smaller the ratio of the adjacent side over the hypotenuse. Ask pupils to explain why the cosine of an angle between 0° and 90° will always be between 0 and 1. 54 0.588 0.586 0.585 0.584 0.582 0.581 55 0.574 0.572 0.571 0.569 0.568 0.566 56 0.559 0.558 0.556 0.555 0.553 0.552

The cosine ratio using a calculator What is the value of cos 25°? To find the value of cos 25° using a scientific calculator, start by making sure that the calculator is set to work in degrees. Key in: cos 2 5 = Your calculator should display 0.906307787 This is 0.906 to 3 significant figures. Some calculators require you to put in the angle first: 65 cos. In this case you do not need the equals sign.

The tangent ratio the length of the opposite side The ratio of is the tangent ratio. the length of the adjacent side The value of the tangent ratio depends on the size of the angles in the triangle. O P S I T E opposite tan θ = adjacent θ A D J A C E N T

The tangent ratio What is the value of tan 71°? 11.6 cm 4 cm This is the same as asking: In a right-angled triangle with an angle of 71°, what is the ratio of the opposite side to the adjacent side? To work this out, an accurately drawn right-angled triangle with a 71° angle could be drawn and the lengths of the opposite side and the adjacent side measured.

The tangent ratio What is the value of tan 71°? It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 71° are similar. The length of the opposite side divided by the length of the adjacent side will always be the same value as long as the angle is the same. In this triangle, 71° 11.6 cm 4 cm opposite Teacher notes Notice that the tangent ratio differs from the sine and cosine ratios in that it can be larger than 1. This is because when the angle we are concerned with is greater than 45°, the side opposite that angle will be longer than the side adjacent to the angle. When we divide a larger number by a smaller number the answer is always greater than 1. tan 71° = adjacent 11.6 4 = = 2.9

The tangent ratio using a table What is the value of tan 71°? Here is an extract from a table of tangent values: Angle in degrees 0.0 0.1 0.2 0.3 0.4 0.5 70 2.75 2.76 2.78 2.79 2.81 2.82 71 2.90 2.92 2.94 2.95 2.97 2.99 72 3.08 3.10 3.11 3.13 3.15 3.17 Teacher notes Discuss how to find the value of tan 71° using the table. Discuss how other values can be found from the table. Ask pupils If the side opposite an acute angle in a right-angled triangle is 4 times longer than the side adjacent to the angle, how big is the angle? From the table the angle is about 76°. Remind pupils that these numbers tell us how many times bigger the side opposite the angle is than the side adjacent to the angle. 73 3.27 3.29 3.31 3.33 3.35 3.38 74 3.49 3.51 3.53 3.56 3.58 3.61 75 3.73 3.76 3.78 3.81 3.84 3.87 76 4.01 4.04 4.07 4.10 4.13 4.17

The tangent ratio using a calculator What is the value of tan 71°? To find the value of tan 71° using a scientific calculator, start by making sure the calculator is set to work in degrees. Key in: tan 7 1 = Your calculator should display 2.904210878 This is 2.90 to 3 significant figures. Some calculators require you to put in the angle first: 71 tan.

Match the equivalents Teacher notes Ask pupils to match the equivalents and then state what the ratios actually are.

The three trigonometric ratios θ O P S I T E H Y N U A D J A C E N T Opposite S O H Sin θ = Hypotenuse Adjacent C A H Cos θ = Hypotenuse Teacher notes Stress to pupils that they must learn these three trigonometric ratios. Pupils can remember these using SOHCAHTOA or they may wish to make up their own mnemonics using these letters. Opposite T O A Tan θ = Adjacent

The relationship between sine and cosine The sine of a given angle is equal to the cosine of the complement of that angle. This can be written as: sin θ = cos (90 – θ) This can be shown as: a b 90 – θ θ a b sin θ = a b cos (90 – θ) = a b