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© T Madas © T Madas We know that: Opp tanq = Adj it is also true that: Opp Adj = tanq and: Opp = tanq ´ Adj How can I possibly remember all that? What do I need it for anyway? © T Madas Remember T. O. A. Opposite Tangent x Adjacent © T Madas Opposite How tall is the tower? Adjacent θ © T Madas Opposite How tall is the tower? Adjacent = 40 m 37° © T Madas How tall is the tower? O T A Opposite Opposite Tangent x Adjacent Adjacent = 40 m 37° © T Madas How tall Opp is the tower? = Adj x tanθ Opp = 40 x tan37° Opposite Opp = 30.14 m Adjacent = 40 m 37° © T Madas More Tangent Calculations O T A ? 30° 10 m Opposite Tangent x Adjacent Opp = Adj x tanθ Opp = 10 x tan30° Opp = 5.77 m © T Madas More Tangent Calculations 18 m O T A Opposite ? 40° Tangent x Adjacent Adj = Opp ÷ tanθ Adj = 18 ÷ tan40° Adj = 21.45 m © T Madas The Sine The Cosine Opposite Adjacent Sine x Hypotenuse Cosine x Hypotenuse Opp sin q = Hyp Adj cos q = Hyp S O H C A H --- The Tangent Opposite Tangent x Adjacent tan q = --- T O Opp Adj A © T Madas © T Madas Sine & Cosine Calculations 15 m O S H 40° Opposite Sine x Hypotenuse Hyp = Opp ÷ sinθ Hyp = 15 ÷ sin40° Hyp = 23.34 m © T Madas Sine & Cosine Calculations A C H 30° 10 m Adjacent Cosine x Hypotenuse Hyp = Adj ÷ cosθ Hyp = 10 ÷ cos30° Hyp = 11.55 m © T Madas Sine & Cosine Calculations A C H ? 27° Adjacent Cosine x Hypotenuse Adj = cosθ x Hyp Adj = cos27° x 12 Adj = 10.69 m © T Madas Sine & Cosine Calculations O S H ? 21° Opposite Sine x Hypotenuse Opp = sinθ x Hyp Opp = sin21° x 15 Opp = 5.38 m © T Madas © T Madas A cylindrical glass has a diameter of 6 cm and a straw is resting on the glass as shown, the straw protruding above the rim of the glass by 5 cm. The straw forms an angle of 65° with the base of the glass. 1. Calculate the height of glass, correct to 1 decimal place. 2. Calculate the length of the straw, correct to 1 decimal place. 12.9 cm x = tan65° x 6 x c SOH CAH TOA x ≈ 12.9 cm 65° 6 cm © T Madas A cylindrical glass has a diameter of 6 cm and a straw is resting on the glass as shown, the straw protruding above the rim of the glass by 5 cm. The straw forms an angle of 65° with the base of the glass. 1. Calculate the height of glass, correct to 1 decimal place. 2. Calculate the length of the straw, correct to 1 decimal place. y 65° 12.9 cm We can find y by using Pythagoras Theorem or trigonometry. If we use Pythagoras Theorem we must obtain the value of 12.9 to greater accuracy. We are going to use trigonometry instead so we can use the length of 6 cm. 6 cm © T Madas A cylindrical glass has a diameter of 6 cm and a straw is resting on the glass as shown, the straw protruding above the rim of the glass by 5 cm. The straw forms an angle of 65° with the base of the glass. 1. Calculate the height of glass, correct to 1 decimal place. 2. Calculate the length of the straw, correct to 1 decimal place. SOH CAH TOA y 12.9 cm 6 y = cos65° c The straw is 19.2 cm long y ≈ 14.2 cm 65° 6 cm © T Madas © T Madas The diagram below shows an alley between two houses. A ladder 3.6 m long is placed against house B as shown. The angle between the ground and the ladder is 70°. Calculate how wide is the alley, giving your answer correct to 3 significant figures. B c A x = cos70° x 3.6 x ≈ 0.342 x 3.6 c SOH CAH TOA x = 1.23 m [3 s.f.] 70° x 1.23 m © T Madas The diagram below shows an alley between two houses. A different ladder is placed against house A as shown. This ladder is 4.2 m long. Calculate the angle the ladder makes with the wall, giving your answer correct to the nearest degree. 4.2 A B = sinθ sinθ ≈ 0.293 θ = sin-1[0.293] 1.23 m c θ c 1.23 c SOH CAH TOA θ = 17° [nearest degree] © T Madas © T Madas A 43681 + 88209 = d 2 d 131890 = d 2 297 mm D 131890 = d c 2092 + 2972 = d 2 c C c 209 mm B c An A4 size sheet of card is a rectangle 297 mm long by 209 mm wide. (a) Calculate the length of its diagonal, to 3 s.f. (b) Find the acute angle formed by the diagonal and the longer of the two sides of the rectangle, to 2 s.f. d ≈ 363 mm © T Madas An A4 size sheet of card is a rectangle 297 mm long by 209 mm wide. (a) Calculate the length of its diagonal, to 3 s.f. (b) Find the acute angle formed by the diagonal and the longer of the two sides of the rectangle, to 2 s.f. B C Which trig ratio can we use? 209 mm A We can use trig to find θ θ 297 mm D Why should we avoid using a ratio which involves the length of 363 mm? © T Madas An A4 size sheet of card is a rectangle 297 mm long by 209 mm wide. (a) Calculate the length of its diagonal, to 3 s.f. (b) Find the acute angle formed by the diagonal and the longer of the two sides of the rectangle, to 2 s.f. A Opp tanθ = adj c 209 mm C 35° θ 297 mm SOH CAH TOA D θ = tan-1 c 209 tanθ = 297 209 297 c B θ ≈ 35° © T Madas © T Madas ABCD is a quadrilateral with RDCB = RADB = 90°, RDAB = 40°, DB = 9 cm and CB = 7 cm. Calculate to 3 significant figures: 1. 2. The length of CD The length of AB 3. The size of RCBD A 72 + x2 = 92 D C 7 cm B c x 2 = 32 c x x 2 = 81 – 49 x = 32 c 5.66 cm 49 + x 2 = 81 c 41° c By Pythagoras Theorem: x ≈ 5.66 cm [3 s.f.] © T Madas ABCD is a quadrilateral with RDCB = RADB = 90°, RDAB = 40°, DB = 9 cm and CB = 7 cm. Calculate to 3 significant figures: 1. 2. The length of CD The length of AB 3. The size of RCBD A SOH CAH TOA 5.66 cm y C 9 sin41° c D y = y = 9 0.656 c 41° y = 13.7 cm [3 s.f.] 7 cm B © T Madas ABCD is a quadrilateral with RDCB = RADB = 90°, RDAB = 40°, DB = 9 cm and CB = 7 cm. Calculate to 3 significant figures: 1. 2. The length of CD The length of AB 3. The size of RCBD Why should we avoid using a trig ratio which involves the length CD ? A C 7 9 cosθ ≈ 0.777 θ 7 cm θ = cos-1[0.777 ] B c 5.66 cm cosθ = c D c SOH CAH TOA 41° θ = 38.9° [3 s.f.] © T Madas © T Madas The figure below shows the cross section of a barn. AE = AB and all lengths are in metres [not to scale] 1. Calculate the length of AB. 2. Calculate the size of the angle ABE, giving your answer to the nearest degree. 11 D 9 C 7.84 + 20.25 = x 2 c B 8.2 4.5 2.82 + 4.52 = x 2 c x x 2 = 28.09 c E By Pythagoras Theorem: x = 28.09 c 2.8 A x = 5.3 m lengths in metres © T Madas The figure below shows the cross section of a barn. AE = AB and all lengths are in metres [not to scale] 1. Calculate the length of AB. 2. Calculate the size of the angle ABE, giving your answer to the nearest degree. Since we know the exact values of all the sides of the right angled triangle we can use any one of the three trigonometric ratios: A D 9 lengths in metres C sinθ = 2.8 c 11 8.2 4.5 B 5.3 sinθ ≈ 0.528 θ = sin-1[0.528] c θ E c 2.8 SOH CAH TOA θ = 32° [nearest degree] © T Madas © T Madas The figure below shows a right angled trapezium ABCD. All lengths are in cm [not to scale] 1. Calculate the length of BC. 2. Calculate the size of the angle DCB, giving your answer correct to 1 decimal place. 3.3 lengths in cm 10.89 + 31.36 = x 2 C c c x x 2 = 42.25 c 6 3.32 + 5.62 = x 2 x = 42.25 c D By Pythagoras Theorem: B 2.7 5.6 5.6 A x = 6.5 cm © T Madas The figure below shows a right angled trapezium ABCD. All lengths are in cm [not to scale] 1. Calculate the length of BC. 2. Calculate the size of the angle DCB, giving your answer correct to 1 decimal place. 6 3.3 lengths in cm C tanθ = 5.6 3.3 tanθ ≈ 1.697 θ = tan-1[1.697 ] c θ c SOH CAH TOA c D B 2.7 5.6 5.6 A Since we know the exact values of all the sides of the right angled triangle we can use any one of the three trigonometric ratios: θ = 59.5° [1 d.p.] © T Madas © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: 1. The vertical height of B above the level of A 2. The vertical height of D above the level of A 3. The length EC 4. The height of the water level if the tank was to be stood upright with AD horizontal. C E B D 40° A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: 1. The vertical height of B above the level of A C SOH CAH TOA x = 40 x sin40° x = 25.7 cm [3 s.f.] E B D x 40° A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: 1. The vertical height of B above the level of A 2. The vertical height of D above the level of A 3. The length EC 4. The height of the water level if the tank was to be stood upright with AD horizontal. C E B D x 40° A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: 1. The vertical height of B above the level of A 2. The vertical height of D above the level of A C SOH CAH TOA y = 30 x sin50° y = 23.0 cm [3 s.f.] E B D x y 50° 40° A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: 1. The vertical height of B above the level of A 2. The vertical height of D above the level of A 3. The length EC 4. The height of the water level if the tank was to be stood upright with AD horizontal. C α E 40° B D x y 50° 40° A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: C SOH CAH TOA 30 α = 0.839 c 30 tan40° E 40° B c α = α α = 35.8 cm [3 s.f.] D x y 50° 40° A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: 1. The vertical height of B above the level of A 2. The vertical height of D above the level of A 3. The length EC 4. The height of the water level if the tank was to be stood upright with AD horizontal. C E B D A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: C E B D A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: C B E D A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: C B E D A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: C B E D A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: C B E D A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: The area of the rectangle (above the water level) must be equal to the area of the triangle BCE C B h 1 x 30 x 35.8 = 30 x h 2 h = 17.9 cm E D A © T Madas The figure below shows the rectangular cross-section ABCD of a fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side AB tilted at an angle of 40° to the horizontal. The water inside the tank is level with point B. Calculate to 3 significant figures: 1. The vertical height of B above the level of A 2. The vertical height of CD above the level of A 3. The length EC 4. The height of the water level if the tank was to be stood upright with AD horizontal. B h E D A © T Madas © T Madas