Trig functions in a right triangle

 Soh Cah Toa

One thing that stands out from my high school geometry class 100 years ago is

 Soh Cah Toa . Which one reads as:

the Sin (sine) of an angle is the  opposite side over the hypotenuse,

the Cos (cosine) of an angle is the adjacent side over the hypotenuse, and

the Tan (tangent) is the opposite side over the adjacent side.

This discussion applies to right triangles only. By definition, a triangle has 3 sides. To name the side opposite angle A, just go from the vertex A, out the middle of the angle (see the dashed line in the diagram below) till you hit the side opposite it, which is side BC or alternatively, side a. The hypotenuse is always the side opposite the right angle. If C is the right angle, then AB or c, is the hypotenuse. The third side must be the adjacent side, AC or b. Notice that the side opposite angle B is b and the side adjacent to angle B is a. the hypotenuse remains AB or c.

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    Notice that   sin A = cos B . The word cosine comes from the complement of the angle. Two angles are complementary if the add to 90' and in the case above  A + B =  90'. If A = 20'  and B = 70', then sin 20' = cos 70', and the sin 40' = cos 50', and sin A' = cos (90'-A).

2. Solve for x:     

     Given: a right triangle, the hypotenuse is 10 m, the angle shown is 30'

      sin 30' = x/10   (sine = opp/hypotenuse) , then multiply both sides by 10 to get

      x = 10*sin 30'   On your calculator enter 10*sin(30) then ENTER (make sure the angle  is in degrees rather than radians).

       x = 5 m   ANSWER

 

3.  Find the sine, cosine, and tangent of angles L and N. Use Pythagoras to find LN first

 

LN2 = 82 + 152 = 64 + 225 = 289

LN2 = 289

LN = Sqrt(289) = 17

 

            sin L = 8/17                       sin N =  15/17

 

 

            cos L = 15/17                    cos N = 8/17   

 

 

             tan L = 8/15                       tan N = 15/8

 

To find angle L, use the calculator

angle L = sin-1(8/17) ENTER   28' answer

                     'sin-1(8/15)' is read as, 'the angle whose sine is 8/15' (which equals 28')

OR

To find angle L,  use the calculator

angle L = cos-1(15/17) ENTER   28' answer

 

To find angle N,  use the calculator

                             angle N = tan-1(15/8) ENTER   62' answer