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© Boardworks Ltd 2005 1 of 43 S3 Trigonometry KS4 Mathematics

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© Boardworks Ltd 2005 2 of 43 A A A A A A S3.1 Right-angled triangles S3 Trigonometry Contents S3.2 The three trigonometric ratios S3.3 Finding side lengths S3.4 Finding angles S3.5 Angles of elevation and depression S3.6 Trigonometry in 3-D

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© Boardworks Ltd 2005 3 of 43 Right-angled triangles A right-angled triangle contains a right angle. The longest side opposite the right angle is called the hypotenuse.

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© Boardworks Ltd 2005 4 of 43 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. The side between the marked angle and the right angle is called the adjacent side. x

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© Boardworks Ltd 2005 5 of 43 Label the sides

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© Boardworks Ltd 2005 6 of 43 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. The ratio of the side lengths in each triangle is the same. 3 4 = 6 8 opp adj = 4 5 = 8 10 adj hyp = 3 5 = 6 10 opp hyp = 3 cm 4 cm 5 cm 37° 8 cm 6 cm 10 cm 37°

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© Boardworks Ltd 2005 7 of 43 Similar right-angled triangles

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© Boardworks Ltd 2005 8 of 43 A A A A A A S3.2 The three trigonometric ratios Contents S3 Trigonometry S3.3 Finding side lengths S3.4 Finding angles S3.5 Angles of elevation and depression S3.6 Trigonometry in 3-D S3.1 Right-angled triangles

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© Boardworks Ltd 2005 9 of 43 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. 30° 8 cm ? 4 cm 6 cm 12 cm ? 30°

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© Boardworks Ltd 2005 10 of 43 The sine ratio The ratio of the length of the opposite side the length of the hypotenuse is the sine ratio. The value of the sine ratio depends on the size of the angles in the triangle. θ OPPOSITEOPPOSITE H Y P O T E N U S E We say: sin θ = opposite hypotenuse

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© Boardworks Ltd 2005 11 of 43 The sine ratio What is the value of sin 65°? 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. 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:

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© Boardworks Ltd 2005 12 of 43 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, sin 65° = opposite hypotenuse 65° 10 cm 11 cm = 10 11 = 0.91 (to 2 d.p.)

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© Boardworks Ltd 2005 13 of 43 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 63 64 65 66.0 0.891 0.899 0.906 0.914.1 0.892 0.900 0.907 0.914.2 0.893 0.900 0.908 0.915.3 0.893 0.901 0.909 0.916.4 0.894 0.902 0.909 0.916.5 0.895 0.903 0.910 0.917

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© Boardworks Ltd 2005 14 of 43 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 65= Your calculator should display 0.906307787 This is 0.906 to 3 significant figures.

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© Boardworks Ltd 2005 15 of 43 The cosine ratio The ratio of the length of the adjacent side the length of the hypotenuse 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 E

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© Boardworks Ltd 2005 16 of 43 The cosine ratio 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. What is the value of cos 53°? This is the same as asking:

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© Boardworks Ltd 2005 17 of 43 The cosine ratio 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, cos 53° = adjacent hypotenuse 53° 6 cm 10 cm = 6 10 = 0.6 What is the value of cos 53°?

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© Boardworks Ltd 2005 18 of 43 The cosine ratio using a table What is the value of cos 53°? 0.588 Here is an extract from a table of cosine values: Angle in degrees 50 51 52 53.0 0.643 0.629 0.616 0.602.1 0.641 0.628 0.614 0.600.2 0.640 0.627 0.613 0.599.3 0.639 0.625 0.612 0.598.4 0.637 0.624 0.610 0.596.5 0.636 0.623 0.609 0.595 54 55 56 0.574 0.559 0.586 0.572 0.558 0.585 0.571 0.556 0.584 0.569 0.555 0.582 0.568 0.553 0.581 0.566 0.552

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© Boardworks Ltd 2005 19 of 43 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 25= Your calculator should display 0.906307787 This is 0.906 to 3 significant figures.

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© Boardworks Ltd 2005 20 of 43 The tangent ratio The ratio of the length of the opposite side the length of the adjacent side is the tangent ratio. The value of the tangent ratio depends on the size of the angles in the triangle. θ OPPOSITEOPPOSITE We say, tan θ = opposite adjacent A D J A C E N T

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© Boardworks Ltd 2005 21 of 43 The tangent ratio What is the value of tan 71°? 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 we can accurately draw a right-angled triangle with a 71° angle and measure the lengths of the opposite side and the adjacent side. This is the same as asking:

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© Boardworks Ltd 2005 22 of 43 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, tan 71° = opposite adjacent 71° 11.6 cm 4 cm = 11.6 4 = 2.9

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© Boardworks Ltd 2005 23 of 43 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 70 71 72 73.0 2.75 2.90 3.08 3.27.1 2.76 2.92 3.10 3.29.2 2.78 2.94 3.11 3.31.3 2.79 2.95 3.13 3.33.4 2.81 2.97 3.15 3.35.5 2.82 2.99 3.17 3.38 74 75 76 3.49 3.73 4.01 3.51 3.76 4.04 3.53 3.78 4.07 3.56 3.81 4.10 3.58 3.84 4.13 3.61 3.87 4.17

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© Boardworks Ltd 2005 24 of 43 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 71= Your calculator should display 2.904210878 This is 2.90 to 3 significant figures.

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© Boardworks Ltd 2005 25 of 43 Calculate the following ratios Use your calculator to find the following to 3 significant figures. 1) sin 79° =0.9822) cos 28° =0.883 3) tan 65° =2.144) cos 11° =0.982 5) sin 34° =0.5596) tan 84° =9.51 7) tan 49° =1.158) sin 62° =0.883 9) tan 6° =0.10510) cos = 0.55956°

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© Boardworks Ltd 2005 26 of 43 The relationship between sine and cosine The sine of a given angle is equal to the cosine of the complement of that angle. We can write this as, sin θ = cos (90 – θ ) We can show this as follows, θ a b sin θ = a b a b cos (90 – θ ) = a b 90 – θ

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© Boardworks Ltd 2005 27 of 43 θ OPPOSITEOPPOSITE H Y P O T E N U S E A D J A C E N T The three trigonometric ratios Sin θ = Opposite Hypotenuse S O H Cos θ = Adjacent Hypotenuse C A H Tan θ = Opposite Adjacent T O A Remember: S O H C A H T O A

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© Boardworks Ltd 2005 28 of 43 A A A A A A S3.3 Finding side lengths Contents S3 Trigonometry S3.4 Finding angles S3.5 Angles of elevation and depression S3.6 Trigonometry in 3-D S3.2 The three trigonometric ratios S3.1 Right-angled triangles

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© Boardworks Ltd 2005 29 of 43 Finding side lengths If we are given one side and one acute angle in a right-angled triangle we can use one of the three trigonometric ratios to find the lengths of other sides. For example, 56° x 12 cm Find x to 2 decimal places. We are given the hypotenuse and we want to find the length of the side opposite the angle, so we use: sin θ = opposite hypotenuse sin 56° = x 12 x = 12 × sin 56° = 9.95 cm

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© Boardworks Ltd 2005 30 of 43 Finding side lengths A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground. 5 m 70° x What is the distance between the base of the ladder and the wall? We are given the hypotenuse and we want to find the length of the side adjacent to the angle, so we use: cos θ = adjacent hypotenuse cos 70° = x 5 x = 5 × cos 70° = 1.71 m (to 2 d.p.)

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© Boardworks Ltd 2005 31 of 43 Finding side lengths

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© Boardworks Ltd 2005 32 of 43 A A A A A A S3.4 Finding angles Contents S3 Trigonometry S3.5 Angles of elevation and depression S3.6 Trigonometry in 3-D S3.2 The three trigonometric ratios S3.1 Right-angled triangles S3.3 Finding side lengths

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© Boardworks Ltd 2005 33 of 43 The inverse of sin sin θ = 0.5, what is the value of θ ? To work this out use the sin –1 key on the calculator. sin –1 0.5 =30° sin –1 is the inverse of sin. It is sometimes called arcsin. 30° 0.5 sin sin –1

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© Boardworks Ltd 2005 34 of 43 The inverse of cos Cos θ = 0.5, what is the value of θ ? To work this out use the cos –1 key on the calculator. cos –1 0.5 =60° Cos –1 is the inverse of cos. It is sometimes called arccos. 60° 0.5 cos cos –1

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© Boardworks Ltd 2005 35 of 43 The inverse of tan 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. 45° 1 tan tan –1

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© Boardworks Ltd 2005 36 of 43 Finding angles We are given the lengths of the sides opposite and adjacent to the angle, so we use: tan θ = opposite adjacent tan θ = 8 5 = 57.99° (to 2 d.p.) θ 5 cm 8 cm θ = tan –1 (8 ÷ 5) Find θ to 2 decimal places.

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© Boardworks Ltd 2005 37 of 43 Finding angles

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© Boardworks Ltd 2005 38 of 43 A A A A A A S3.5 Angles of elevation and depression Contents S3 Trigonometry S3.6 Trigonometry in 3-D S3.4 Finding angles S3.2 The three trigonometric ratios S3.1 Right-angled triangles S3.3 Finding side lengths

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© Boardworks Ltd 2005 39 of 43 Angles of elevation

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© Boardworks Ltd 2005 40 of 43 Angles of depression

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© Boardworks Ltd 2005 41 of 43 A A A A A A S3.6 Trigonometry in 3-D Contents S3 Trigonometry S3.5 Angles of elevation and depression S3.4 Finding angles S3.2 The three trigonometric ratios S3.1 Right-angled triangles S3.3 Finding side lengths

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© Boardworks Ltd 2005 42 of 43 Angles in a cuboid

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© Boardworks Ltd 2005 43 of 43 Lengths in a square-based pyramid

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