Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Introduction To Trigonometry. h 37 o 28m Calculate the height of the tree: opp tan x adj o ………………. What Does Trigonometry Do ? Without trigonometry we wouldn’t have sailed the world , satellites in space wouldn’t know where they were and planes would be lost. Without trigonometry architects couldn’t design buildings and engineers couldn’t make cars, planes….. Without trigonometry millions of calculations made all over the world to keep things moving couldn’t be made. Naming Sides Of Triangles. Consider the right angled triangle shown below: Hyp’ Opp’ xo Adj’ It has an angle of size xo marked in the right hand corner. The side opposite the angle you are looking at is called the opposite side. Opp’ for short. The side opposite the right angle is called the hypotenuse. Hyp’ for short. The remaining side is called the adjacent. Adj’ for short. The Ratio Of Two Sides. Look at the right angled triangle ABC below: If AB = BC = 6cm what type of triangle is ABC. Isosceles. Calculate the ratio of opp’ adj’ for angle xo . Opp’ = 6 C Opp’ = 6 Adj, 6 6cm Opp’ xo A Adj’ 6cm B Adj’ = 6 = 1 What size is must the angle xo ? xo = 45o Now consider the same calculation again for the triangle below: We can see at once that the opposite divided by the adjacent is 1 and that the angle xo is still 45o C The same result will occur for all Right angled isosceles triangles. 10cm xo A 10cm B The Tangent Of An Angle. C The ratio of opposite divided by adjacent is called the tangent of the angle xo . Opp’ Normally written as tan xo for short. xo A Adj’ B Key Result. opp tan x adj o For all right angled triangles the value of the tangent ratio will determine the size of the angle xo regardless of the size of the triangle. Using The Tangent Ratio. Consider once more the triangle we started with: C To calculate xo on a calculator follow the steps below: 6cm tan x o Opp’ 6 tan x 6 xo o A Adj’ 6cm B On your calculator select the following buttons: INV Tan-1 opp adj 1 = tan x o 1 x o tan 1 1 Xo = 45o Calculating Angles Using The Tangent Ratio. We have now found how to calculate angles using the tangent ratio as the following example shows. Calculate the angle ao below. adj’ 11cm ao (1) Identify the opposite side. (2) Identify the adjacent side. 16cm opp’ opp tan x adj (3) Write down the tangent ratio. o tan x o 16 11 tan x o 1.455 x o tan 1 1.455 xo = 55.5o (4) Substitute your values . (5) Divide the ratio giving your answer to 3d.p. (6) Use your calculator to find the angle ao to 1d.p. Further Example. 12m Opp’ adj’ Calculate the angle bo. tan x o 16m bo opp adj 12 tan x 16 o tan x o 0.75 x o tan 1 0.75 xo = 36.9 o Always follow the routine !! What Goes In The Box ? Find the size of each of the unknown angles below using the tangent ratio. ans: 37.9 o (1) ans: (2) 51.5 o bo 14cm 27cm ao 18cm (3) 2.7cm c o ans: 33.7 o 34cm 34.7m (3) do 29.1m ans: 40 o 1.8cm