Trigonometry-4 Ratios of the Sides of Triangles. The meaning of sin, cos and tan sin is short for sine cos is short for cosine tan is short for tangent

Trigonometry Triangles – Ratios of Sides Construct a Right Angled Triangle Ratios in a Right Angled Triangle

Triangles – Ratios of Sides  In triangle ABC, AB = 5 units, BC = 3 units and AC = 4 units.  A ratio is one number divided by another number. BC A B C  The ratio of side BC to side AC is written as: AC4 3 = 0,75 =

Triangles – Ratios of Sides  In triangle ABC you are given the length of the sides.  A ratio is one number divided by another number. BC A B C  The ratio of side BC to side AB is written as: AB5 3 = 0,6 =

Triangles – Ratios of Sides  In triangle ABC you are given the length of the sides.  A ratio is one number divided by another number. AC A B C  The ratio of side AC to side AB is written as: AB5 4 = 0,8 =

Construct a Right Angled Triangle The tools you need: ruler pencil protractor set square

Construct a Right Angled Triangle with an angle of 30  1: Draw a line

Construct a Right Angled Triangle 1: Draw a line

Construct a Right Angled Triangle 2: Make a point on the line

Construct a Right Angled Triangle 3: Move the protractor onto the point

Construct a Right Angled Triangle 4: Mark your angle with a point

Construct a Right Angled Triangle 4: Mark your angle with a point

Construct a Right Angled Triangle 5: Draw a line through the two points

Construct a Right Angled Triangle 5: Draw a line through the two points

Construct a Right Angled Triangle 6: Now draw a right angle to complete the triangle.

Construct a Right Angled Triangle 6: Now draw a right angle to complete the triangle.

Construct a Right Angled Triangle 30  7: Write in the size of the angle.

Ratios in a Right Angled Triangle Label the hypotenuse H, the opposite side O and the adjacent side A. 30  H O A

Ratios in a Right Angled Triangle Measure the length of H, O and A. 30  H O A Do you think everyone will get the same answers? No – all the triangles are different. The only thing that is the same is the angle.

Ratios in a Right Angled Triangle Measure the length of H, O and A. 30  H O A A = H = O = 5,0 Remember – your answers won’t be the same.

Ratios in a Right Angled Triangle Measure the length of H, O and A. 30  H O A A = H = O = 5,0 5,7 Remember – your answers won’t be the same.

Ratios in a Right Angled Triangle Measure the length of H, O and A. 30  H O A A = H = O = 5,0 5,7 2,8 Remember – your answers won’t be the same.

Ratios in a Right Angled Triangle 30  H O A A = 5,0 H = 5,7 O = 2,8 O A = 2,8 5,0 Work out the ratio: O A =0.56 O A =

Ratios in a Right Angled Triangle 30  H O A A = 5,0 H = 5.7 O = 2,8 O H = 2,8 5,7 Work out the ratio: O H =0.49 O H = O A =0.56

Ratios in a Right Angled Triangle 30  H O A A = 5,0 H = 5.7 O = 2,8 A H = 5.0 5,7 Work out the ratio: A H =0.87 A H = O H =0.49 O A =0.56

Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O A =0.56 O H =0.49 A H =0.87 Do you think our diagrams are very accurate? Probably not! With our tools we can’t be exact! What should we do to our answers?

Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O A =0.56 O H =0.49 A H =0.87 The second decimal place won’t be accurate. Let’s round the answers to one decimal place.

Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O A =0.56 O H =0.49 A H =0.87 How do we round the answers to one decimal place? If the second decimal is 4 or less, then the first decimal stays the same. If the second decimal is 5 or more, then the first decimal becomes bigger by 1.

O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.49 A H =0.87 How do we round the answers to one decimal place? If the second decimal is 5 or more, then the first decimal becomes bigger by 1. If the second decimal is 4 or less, then the first decimal stays the same.

O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.87 How do we round the answers to one decimal place? If the second decimal is 5 or more, then the first decimal becomes bigger by 1. If the second decimal is 4 or less, then the first decimal stays the same.

O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 How do we round the answers to one decimal place? If the second decimal is 5 or more, then the first decimal becomes bigger by 1. If the second decimal is 4 or less, then the first decimal stays the same.

O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 Does everyone in the class get the same answers? YES ! Amazing! From all the different sized triangles you drew we get the same answers for the ratios.

O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 On a scientific calculator do the following: Enter 30. Press the ‘tan’ button. Round the answer to 1 decimal. Answer is same as:

O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 On a scientific calculator do the following: Enter 30. Press the ‘sin’ button. Round the answer to 1 decimal. Answer is same as:

O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 On a scientific calculator do the following: Enter 30. Press the ‘cos’ button. Round the answer to 1 decimal. Answer is same as:

O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 Answers on a scientific calculator: Gives same answer as ‘tan’ button. Gives same answer as ‘sin’ button. Gives same answer as ‘cos’ button.

Ratios in a Right Angled Triangle Instead of an angle of 30  in the triangle, you can put in any angle and get the same results. Use the ‘tan’ button. Use the ‘sin’ button. Use the ‘cos’ button. O A Ratio: O H A H

Ratios in a Right Angled Triangle  Instead of drawing triangles and measuring sides to get the ratios, we can just use a scientific calculator.  In any right angled triangle: O  A H O A Tan  = O H Sin  = A H Cos  =

Ratios in a Right Angled Triangle Tan 20  = 0,36 O A O H A H O 20  A H O 40  A H O 60  A H Tan 40  = 0,83Tan 60  = 1,73 sin 20  = 0,34sin 40  = 0,64sin 60  = 0,86 cos 20  = 0,93cos 40  = 0,76cos 60  = 0,5 These are true for any triangle with these angles.

Ratios in a Right Angled Triangle  We have to remember these trigonometric ratios.  In any right angled triangle: O  A H O A Tan  = O H Sin  = A H Cos  = Some Old Hens Cackle And Howl Till Old Age

Ratios in a Right Angled Triangle O  A H O A Tan  = O H Sin  = A H Cos  = Some Old HensCackle And HowlTill Old Age

Well Done!