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KS4 Mathematics S3 Trigonometry

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**S3.1 Right-angled triangles**

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

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**Right-angled triangles**

A right-angled triangle contains a right angle. The longest side opposite the right angle is called the hypotenuse. Review the definition of a right-angled triangle. 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.

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**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 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 such as x as shown here. Point out that if we labelled the sides with respect to the other acute angle their names would be reversed.

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Label the sides 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.

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**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° Point out that if one acute angle is the same size then all three angles must be the same size 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. Link: N6 Ratio 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

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**Similar right-angled triangles**

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.

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**S3.2 The three trigonometric ratios**

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

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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 ? Stress that no matter how big a right angled triangle is, if one of the angles in 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°

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**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’.

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**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.

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**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.)

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**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 .2 0.893 0.908 0.915 .3 0.901 0.909 0.916 .4 0.894 0.902 .5 0.895 0.903 0.910 0.917 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.

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**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 This is to 3 significant figures.

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**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

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**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.

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**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

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**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 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.

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**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 This is to 3 significant figures.

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**The tangent ratio the length of the opposite side**

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

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**What is the value of tan 71°?**

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? This is the same as asking: 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.

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**What is the value of tan 71°?**

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 tan 71° = opposite adjacent 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. = 11.6 4 = 2.9

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**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 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, about 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.

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**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 This is 2.90 to 3 significant figures.

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**Calculate the following ratios**

Use your calculator to find the following to 3 significant figures. 1) sin 79° = 0.982 2) cos 28° = 0.883 3) tan 65° = 2.14 4) cos 11° = 0.982 5) sin 34° = 0.559 6) tan 84° = 9.51 This exercise provides practice at using of the sin, cos and tan keys on the calculator. It also reviews rounding to a given number of significant figures. Pupils should notice that the sine of a given angle is equal to the cosine of the complement of that angle. They can use this fact to answer question 10 using the answer to question 5. 7) tan 49° = 1.15 8) sin 62° = 0.883 9) tan 6° = 0.105 10) cos = 0.559 56°

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**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, a b cos (90 – θ) = 90 – θ θ a b sin θ =

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**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

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**S3 Trigonometry Contents A S3.1 Right-angled triangles A**

S3.2 The three trigonometric ratios A S3.3 Finding side lengths A S3.4 Finding angles A S3.5 Angles of elevation and depression A S3.6 Trigonometry in 3-D

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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, 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: 56° x 12 cm sin θ = opposite hypotenuse Discuss how we decide which trigonometric ratio to use. sin 56° = x 12 x = 12 × sin 56° = 9.95 cm

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**What is the distance between the base of the ladder and the wall?**

Finding side lengths A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground. 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: 5 m cos θ = adjacent hypotenuse 70° x 5 x cos 70° = x = 5 × cos 70° = 1.71 m (to 2 d.p.)

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Finding side lengths

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**S3 Trigonometry Contents A S3.1 Right-angled triangles A**

S3.2 The three trigonometric ratios A S3.3 Finding side lengths A S3.4 Finding angles A S3.5 Angles of elevation and depression A S3.6 Trigonometry in 3-D

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**The inverse of sin 30° 0.5 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. sin Make sure that pupils can locate the sin–1 key on their calculators. Stress that sin and sin–1 are inverse functions. sin 30° = 0.5 and sin–1 0.5 = 30°. Remind pupils of the use of –1 to denote the multiplicative inverse or reciprocal. This is an extension of this notation. sin–1 30° 0.5

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**The inverse of cos 60° 0.5 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. cos Make sure that pupils can locate the cos–1 key on their calculators. Stress that cos and cos–1 are inverse functions. cos 60° = 0.5 and cos–1 0.5 = 60°. cos–1 60° 0.5

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**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

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**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.)

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Finding angles Use this activity to practice finding the size of angles given two sides in a right-angled triangle.

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**S3.5 Angles of elevation and depression**

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

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Angles of elevation Explain that an angle of elevation is the angle made between the horizon and the top of an object that you are looking up at. Explain that we can use an instrument to measure angles of elevation. By measuring the angle of elevation from the ground to the top of a tall building we can apply trigonometry to determine the height of the building. Hide either the angle of elevation, the height of the building or the distance of the foot of the building to the point of measurement. Ask pupils to use the tangent ratio to find the unknown measurements. Change the position of the observer to change the measurements. Reset the activity to change the building.

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Angles of depression Explain that an angle of depression is the angle made between the horizon and the bottom of an object that you are looking down on. By measuring the angle of depression from the the top of a cliff, for example, we can apply trigonometry to determine the distance of objects from the foot of the cliff. Hide either the angle of depression, the distance between the boat and the viewer or the distance between the boat and the foot of the cliff. Ask pupils to use the tangent ratio to find the unknown measurements. Modify the values by changing the position of the boat.

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**S3 Trigonometry Contents A S3.1 Right-angled triangles A**

S3.2 The three trigonometric ratios A S3.3 Finding side lengths A S3.4 Finding angles A S3.5 Angles of elevation and depression A S3.6 Trigonometry in 3-D

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Angles in a cuboid Stress that when solving problems involving right-angled triangles in 3-D, we should always draw a 2-D diagram of the right-angled triangle in question, labelling all the known sides and angles. Show how angle AGB can be found by first using Pythagoras’ theorem to find the length of BG. Once we know the length of BG we can find angle AGB using the tan ratio. Reveal the answer. If there is a small discrepancy, this is due to rounding. Change the size of the cuboid and ask a volunteer to calculate the new size of angle AGB.

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**Lengths in a square-based pyramid**

Tell pupils that the point M is directly below point E. Point E can be moved to change the angle between the base and the sloping side. Demonstrate how to find the length of EM using the sine ratio. Reveal the answer. If there is a small discrepancy, this is due to rounding. Change the height of the pyramid by dragging on point E and ask a volunteer to calculate the new pyramid height EM. Ask pupils if it is possible to find the length of the side of the square base AB. Challenge pupils to find the length of AB using a combination of trigonometry and Pythagoras’ theorem.

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Trigonometry Right Angled Triangle. Hypotenuse [H]

Trigonometry Right Angled Triangle. Hypotenuse [H]

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