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Trigonometry

Trigonometry. C. Hypotenuse. Opposite. The ‘Hypotenuse’ is always opposite the right angle.  0. F. O. The ‘Opposite’ is always opposite the angle under investigation. Adjacent. C. The ‘Adjacent’ is always alongside the angle under investigation.  0. Hypotenuse. Adjacent. F. O.

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Trigonometry

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  1. Trigonometry C Hypotenuse Opposite The ‘Hypotenuse’ is always opposite the right angle 0 F O The ‘Opposite’ is always opposite the angle under investigation. Adjacent C The ‘Adjacent’ is always alongside the angle under investigation. 0 Hypotenuse Adjacent F O Opposite

  2. Trigonometry C Let us look at the ratio or fraction of the opposite over the adjacent. 3 0 F 6 O Let us now split this triangle up.

  3. Trigonometry C Let us look at the ratio or fraction of the opposite over the adjacent B 3 2 0 4 E F 2 O Let us split it again.

  4. This ratio of is the same for all of the similar triangles shown here. Trigonometry C Let us look at the ratio or fraction of the opposite over the adjacent B 3 A 2 1 0 E F 2 2 2 O D This ratio is known as the Tangent of 0.

  5. 0 0 O 0 T A adjacent opposite opposite adjacent opposite Memory Aid TOA adjacent

  6. Write down the ratio for tan 0 in each of the following triangles. 2 0 0 5 13 3 5 8 0 12 4

  7. Page 106 Exercise 2 Questions 1 to 6 We can use the tan ratio to find the size of the angle. To do this we use the inverse tan button on our calculators. The tan-1 button This is above the tan button and is activated by pressing the shift, 2nd or inverse key Example. Calculate the size of the angle  in the triangle shown below. 0 5 3 4 Now continue with the rest of the exercise.

  8. Michael wants to measure the height of a tower. He measures the angle of elevation from a spot 20m from the base of the tower and finds it to be 400. 400 Calculate the height of the tower. 20m The tan ratio in action (multiplying both sides by 20)

  9. 1. Sketch the triangle. 2. Label the sides. 3. Show ALL your working. Page 108 Exercise 3A

  10. Calculating the Adjacent Side Find the size of x in the following triangle. 4cm 300 x (multiplying both sides by x) (dividing both sides by tan 300) There is another method !

  11. Find the size of x in the following triangle. 4cm 300 O T A Calculating the Adjacent Side We require the adjacent x Page 110 Exercise 3B

  12. Find the size of 0 in the following triangle. 4cm 0 7 cm Calculating an Angle Page 111 Exercise 4

  13. 4m 3m 0 O S H The Sine of an Angle The ladder will slip if the angle it makes with the ground is less than 400. Can we calculate the angle? The tan ratio will not work as we have 2 unknowns. We do not know the angle or the adjacent. We need a ratio to link the opposite with the hypotenuse. This ratio is known as the SINE ratio. (shortened to sin but pronounced sine) Memory Aid SOH

  14. Write down the ratio for sin 0 in each of the following triangles. 2 0 0 5 13 3 5 8 0 12 4

  15. Page 112 Exercise 5

  16. The sine ratio in action Charlie runs up the 4m long plank to the top of the wall. Calculate his height above the ground when he reaches the top. (multiplying both sides by 4) There is another method !

  17. O S H The sine ratio in action Charlie runs up the 4m long plank to the top of the wall. Calculate his height above the ground when he reaches the top. (We require the opposite) Page 113 Exercise 6A

  18. x 5m 300 O S H Calculating the Hypotenuse Calculate the length of this ladder. (We require the Hypotenuse) Page 114 Exercise 6B

  19. 8m 600 x A C H The Cosine of an Angle For safety this ladder must be set at 600 to the ground. How far from the wall should the foot of the ladder be? The tan and sine ratio will not work as we have 2 unknowns. We do not know the opposite or the adjacent. We need a ratio to link the adjacent with the hypotenuse. This ratio is known as the Cosine ratio. (shortened to cos but pronounced cosine) Memory Aid CAH

  20. 8m 600 x A C H For safety this ladder must be set at 600 to the ground. How far from the wall should the foot of the ladder be? (we require the adjacent) The foot of the ladder must be 4m from the wall. Page 115/116 Exercise 7A and 7B

  21. SOH CAH TOA 4 m h m 180 Which Ratio? So far we have always known which ratio to use. What do we do when we do not? How do we choose the correct ratio? We use this to help us remember: SOH CAH TOA Since we can only solve equations with one unknown, we use the ratio that gives us only one unknown. For example…. Tick what we know and put a question mark for what we want to find………… ? ? The SOH is the only ratio with two things marked so we use the sin ratio.

  22. h m 180 4 m x m SOH SOH CAH CAH TOA TOA 180 4 m Tick what we know and what we want………… ?  ?  The TOA is the only ratio with one unknown so we use the tan ratio. Tick what we know and ? what we want…………   ?  The CAH is the only ratio with one unknown so we use the cos ratio.

  23. x m 350 12 m SOH CAH TOA Choose your ratio and calculate x. ?  ?  The TOA is the only ratio with one unknown so we use the tan ratio.

  24. 24 m x0 12 m SOH CAH TOA Choose your ratio and calculate x.     The CAH is the only ratio with one unknown so we use the cos ratio. MIA S3-3 Ex10.1, p218-219

  25. Find the height of the isosceles triangle shown below. 700 20cm 20cm

  26. Find the height of the isosceles triangle shown below. 700 20cm 20cm

  27. Find the height of the isosceles triangle shown below. 350 350 20cm 20cm

  28. Find the height of the isosceles triangle shown below. 350 350 20cm 20cm

  29. Find the height of the isosceles triangle shown below. 350 20cm 20cm h cm

  30. SOH CAH TOA Find the height of the isosceles triangle shown below.  ? ?  350 20cm 20cm h cm

  31. 1. Sketch the triangle. 2. Label the sides. 3. Choose the correct ratio using SOH CAH TOA MIA S3-3 Ex10.2, p 219-220

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