Starter I think of a number, add 7 and then double it. The answer is 28. What was the original number? I think of another number, subtract 3 and then divide it by 4. My answer is 6. I think of another number, add 4, multiply by 3 and then subtract 4. My answer is 47. What was my original number?

We are Learning to…… Use Angle Facts

Angles Angles are measured in degrees. A quarter turn measures 90°. It is called a right angle. We label a right angle with a small square.

Angles Angles are measured in degrees. A half turn measures 180°. This is a straight line. 180°

Angles Angles are measured in degrees. A three-quarter turn measures 270°. 270°

Angles Angles are measured in degrees. A full turn measures 360°. 360°

Vertically opposite angles When two lines intersect, two pairs of vertically opposite angles are formed. a b c d a = c and b = d Vertically opposite angles are equal.

Angles on a straight line Use this activity to demonstrate that the angles on a straight line always add up to 180°. Hide one of the angles and ask pupils to work out its value. Add another angle to make the problem more difficult.

Angles on a straight line Angles on a line add up to 180. a b This should formally summarize the rule that the pupils deduced using the previous interactive slide. a + b = 180° because there are 180° in a half turn.

Angles around a point Move the points to change the values of the angles. Show that these will always add up to 360º. Hide one of the angles, move the points and ask pupils to calculate the size of the missing angle.

Angles around a point Angles around a point add up to 360. b a c d This should formally summarize the rule that the pupils deduced using the previous interactive slide. a + b + c + d = 360 because there are 360 in a full turn.

Complementary angles When two angles add up to 90° they are called complementary angles. a b Ask pupils to give examples of pairs of complementary angles. For example, 32° and 58º. Give pupils an acute angle and ask them to calculate the complement to this angle. a + b = 90° Angle a and angle b are complementary angles.

Supplementary angles When two angles add up to 180° they are called supplementary angles. b a Ask pupils to give examples of pairs of supplementary angles. For example, 113° and 67º. Give pupils an angle and ask them to calculate the supplement to this angle. a + b = 180° Angle a and angle b are supplementary angles.

Angle Facts Angles at a corner total… 90° Angles on a straight line total… 180° Angles at a point total… 360° When two straight lines cross, the opposite angles… are equal

The Pythagorean Theorem The Pythagorean Theorem states that in a right-angled triangle the square on the hypotenuse is equal to the sum of the squares of the other two sides. c² = a² + b² or a² + b² = c² The HYPOTENUSE is the longest side of the triangle

Find the length of the hypotenuse of a right-angled triangle whose short sides are of length 7cm and 18.3cm (one decimal place) c² = a² + b² c² = 7² + 18.3² c² = 49 + 334.89 c² = 383.89 So… c = 383.89 = 19.59…..cm = 19.6cm to 1dp

Identify the Tangent Ratio We are Learning to…… Identify the Tangent Ratio

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 that your calculator is set to work in degrees. Key in: tan 7 1 = Some calculators require the size of the angle to be keyed in first, followed by the tan key. Your calculator should display 2.904210878 This is 2.90 to 3 significant figures.

Identify the Sine and Cosine Ratios We are Learning to…… Identify the Sine and Cosine Ratios

The sine ratio the length of the opposite side the length of the hypotenuse The ratio of is the sine ratio. The value of the sine ratio depends on the size of the angles in the triangle. θ O P S I T E H Y N U We say: sin θ = opposite hypotenuse 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’.

What is the value of sin 65°? The sine ratio What is the value of sin 65°? In a right-angled triangle with an angle of 65°, what is the ratio of the opposite side to the hypotenuse? This is the same as asking: 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.

What is the value of sin 65°? 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. In this triangle, 65° 10 cm 11 cm sin 65° = opposite hypotenuse This ratio can also be demonstrated using the similar right-angled activity on slide 7. = 10 11 = 0.91 (to 2 d.p.)

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 = Some calculators require the size of the angle to be keyed in first, followed by the sin key. Your calculator should display 0.906307787 This is 0.906 to 3 significant figures.

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

What is the value of cos 53°? The cosine ratio What is the value of cos 53°? In a right-angled triangle with an angle of 53°, what is the ratio of the adjacent side to the hypotenuse? This is the same as asking: 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.

What is the value of cos 53°? 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, 53° 6 cm 10 cm cos 53° = adjacent hypotenuse This ratio can also be demonstrated using the similar right-angled activity on slide 7. = 6 10 = 0.6

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 your calculator is set to work in degrees. Key in: cos 2 5 = Some calculators require the size of the angle to be keyed in first, followed by the cos key. Your calculator should display 0.906307787 This is 0.906 to 3 significant figures.

The three trigonometric ratios θ O P S I T E H Y N U A D J A C E N T Sin θ = Opposite Hypotenuse S O H Cos θ = Adjacent Hypotenuse C A H Tan θ = Opposite Adjacent T O A 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. Remember: S O H C A H T O A

The inverse of tan 45° 1 tan θ = 1, what is the value of θ? To work this out use the tan–1 key on the calculator. tan–1 1 = 45° tan–1 is the inverse of tan. It is sometimes called arctan. tan Make sure that pupils can locate the tan–1 key on their calculators. Stress that tan and tan–1 are inverse functions. tan 45° = 1 and tan–1 1 = 45°. tan–1 45° 1

Finding angles Find θ to 2 decimal places. 5 cm 8 cm Find θ to 2 decimal places. We are given the lengths of the sides opposite and adjacent to the angle, so we use: tan θ = opposite adjacent On the calculator we can key in tan–1 (8 ÷ 5). This avoids rounding errors when the ratio cannot be written exactly as a decimal. tan θ = 8 5 θ = tan–1 (8 ÷ 5) = 57.99° (to 2 d.p.)

Solve Equations with Fractions We are Learning to…… Solve Equations with Fractions

= 12 = 5 a² + 6² = 8² 13g = 4² + 7² 3.8² = 6.3² + 3.1² - 3(5.2)(3.8)y x 3 4 x

Homework McGraw – Hill Ryerson 12 Page 72, #s 1 – 10 BLM 2-2 Prerequisite Skills #s 1 - 9