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© Boardworks Ltd 2005 1 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 2 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 3 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 4 of 43 Finding side lengths

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© Boardworks Ltd 2005 5 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 6 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 7 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 8 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 9 of 43 Finding angles

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