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A block of flats is 100m away from a cellphone tower. Someone stands at B . They measure the angle from B up to the top of the tower E to be 62 . This is the angle of elevation. They then measure the angle from B down to the bottom of the tower at C to be 34 . This is the angle of depression.What is the height of the cellph one tower correct to 1 decimal place?

  1. To find the height of the tower, all we have to do is find the length of C D and D E . We see that B D E and B D C are both right-angled triangles. For each of the triangles, we have an angle and we have the length A D . Thus we can calculate the sides of the triangles.

  2. We are given that the length A C is 100m. C A B D is a rectangle so B D = A C = 100 m .

    tan ( C B ^ D ) = C D B D C D = B D × tan ( C B ^ D ) = 100 × tan 34

    Use your calculator to find that tan 34 = 0 , 6745 . Using this, we find that C D = 67 , 45 m

  3. tan ( D B ^ E ) = D E B D D E = B D × tan ( D B ^ E ) = 100 × tan 62 = 188 , 07 m
  4. We have that the height of the tower C E = C D + D E = 67 , 45 m + 188 , 07 m = 255 . 5 m .

Maps and plans

Maps and plans are usually scale drawings. This means that they are an exact copy of the real thing, but are usually smaller. So, only lengths are changed, but all angles are the same. We can use this idea to make use of maps and plans by adding information from the real world.

A ship approaching Cape Town Harbour reaches point A on the map, due south of Pretoria and due east of Cape Town. If the distance from Cape Town to Pretoria is 1000km, use trigonometry to find out how far east the ship is from Cape Town, and hence find the scale of the map.

  1. We already know the distance between Cape Town and A in blocks from the given map (it is 5 blocks). Thus if we work out how many kilometers this same distance is, we can calculate how many kilometers each block represents, and thus we have the scale of the map.

  2. Let us denote Cape Town with C and Pretoria with P . We can see that triangle A P C is a right-angled triangle. Furthermore, we see that the distance A C and distance A P are both 5 blocks. Thus it is an isoceles triangle, and so A C ^ P = A P ^ C = 45 .

  3. C A = C P × cos ( A C ^ P ) = 1000 × cos ( 45 ) = 1000 2 km

    To work out the scale, we see that

    5 blocks = 1000 2 km 1 block = 200 2 km

Mr Nkosi has a garage at his house, and he decides that he wants to add a corrugated iron roof to the side of the garage. The garage is 4m high, and his sheet for the roof is 5m long. If he wants the roof to be at an angle of 5 , how high must he build the wall B D , which is holding up the roof? Give the answer to 2 decimal places.

  1. We see that the triangle A B C is a right-angled triangle. As we have one side and an angle of this triangle, we can calculate A C . The height of the wall is then the height of the garage minus A C .

  2. If B C =5m, and angle A B ^ C = 5 , then

    A C = B C × sin ( A B ^ C ) = 5 × sin 5 = 5 × 0 , 0871 = 0 . 4358 m

    Thus we have that the height of the wall B D = 4 m - 0 . 4358 m = 3 . 56 m .

Applications of trigonometric functions

  1. A boy flying a kite is standing 30 m from a point directly under the kite. If the string to the kite is 50 m long, find the angle ofelevation of the kite.
  2. What is the angle of elevation of the sun when a tree 7,15 m tall casts a shadow 10,1 m long?

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Source:  OpenStax, Basic mathematics review. OpenStax CNX. Jun 06, 2012 Download for free at http://cnx.org/content/col11427/1.2
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