What is trigonometry?

Trigonometry is a branch of geometry that is used to find sides and angles in triangles both right-angled and non right-angled. The first part of the word is from greek trigon "triangle" and the second part is from greek metron "a measure".

Basically trigonometry is all abut triangles

Solving unknown sides

A ship drops its anchor with a 30m cable length to the bottom of the sea, which the anchor makes a 39° with the seabed.  what is the distance? in 2 d.p

sinθ= O/H
sin39= d/30
      d= sin(39)x30
        = 18.88m

Solving angles of elevation & depression

From the top of a light house, Jim sees a rescue boat at an angle of depression of 20° . He knows the lighthouse is 28m high. How far away is the base of the light house is the boat?  correct to 2 decimal places.
















  tan θ= O/A
tan 70 = x/28
        x = tan (70) x 28
           = 76.93m


A woman flies a kite with a 100m  long string. The angle of elevation of the string is 52°. How high off the ground is the kite to the nearest metre?

     sin θ= O/H
   sin52= x/100
         x= sin (52) x 100
           = 79m 

Solving bearings

A speed boat travels 10km south then 8km east. Find the speed boats bearing from its starting point to the nearest degree.


















     tanθ= (8/10)
          θ= tan^-1(8/10)
            = 39°              (nearest degree)

The bearing from B to O= 180°-θ   
                                    = 180° -39°                                     
                                    = 141° 



A family on a road trip drives 20km on a bearing of 330° from point A to point B, Find how far west point B is from point A.

















sin30= x/20
      x= 20sin30
        = 10km

A pigeon delivers a letter from its master on a bearing of 315°The pigeon is flying at a speed of 50km/hour. How far west of the castle is the pigeon? to one d.p

cos45= x/50
       x= 50cos45
         = 35.4km

Pythagoras' theorem

SOHCAHTOA

Bearings

A Bearing is used to represent the direction of one point relative to another point. The bearing of a point is the number of degrees in the angle measured in a clockwise direction from the North line to the line joining the centre of the compass with the point.

For example, the bearing of A from B is 065º
The Bearing of B from A is 245º

Note:
- Three figures are used to give bearings
- All Bearings are measured in a horizontal plane
- measured clockwise from north

Finding unknown sides

We can find an unknown side in a right-angle triangle when we know:

-  one length

-  one angle (apart from the right angle)

  Tan40° = O/A

                 =  7/x

 xTan40° = 7

               x= 7/Tan40°               

               x= 8°

Finding angles

When we need to find an unknown angle in a right- angle triangle, we must use the inverse of that particular function.











Step 1: find the names of the two sides you know

Step 2: now use the first letters of those sides,  (Opposite and Hypotenuse) and phrase (SOHCAHTOA)

Step 3: O and H are present in SOH, which tells us we need to use sine into the equation, then we must use the inverse, this can be found by pressing "shift" then sin on the calculator.

            sinθ= 5/13
             θ= sin^-1(5/13)
               = 23°

Angles of elevation and depression

The angle of elevation is the angle between a horizontal line and the line of sight from the observer to an object at a high level.

The angle of depression is the angle between a horizontal line of sight from he observer to an object at a lower level.

Applications in sport

Baseball

Use the law of cosines to determine how far first base is from the pitcher's rubber. 

Soccer

Basketball

If this guy shot the basketball in the hoop, what would be the height at which the player shot if he was 15ft from the basketball hoop and the height of the hoop was 10ft?

The solution is to use Pythagoras' theorem, then to square root the answer that you got to get the height at which he shot

c2= a2+b2

x2=152+102

x2=225+100

x=18 ft

Swimming

Kristin is swimming in the ocean and notices a coral reef below her. The angle of depression is 35°and the depth of the ocean, at that point is 250 feet. How far away is she from the reef? 

   Sinθ= O/H

  sin35= x/250

           x= sin(35)x250

           x= 143ft