© 2008  Rasmus ehf
and Jóhann Ísak

Trigonometry sine, cosine and tangent.

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Lesson 1

ABC is a right angled triangle

The angle  A  is  30 degrees. We write this as:

 

a    is the symbol for the side opposite angle A

b    is the symbol for the side opposite angle B

c    is the symbol for the side opposite angle C

Similar triangles are triangles in which all the angles in one triangle are equal to the angles in the other triangle

These two triangles are similar. The ratio between two sides in one triangle is equal to the ratio between the corresponding sides in the other triangle.

Using the notation in the above triangles we get the following:

                  

The ratio depends on the size of the angle.

Tangent

The ratio called tangent (tan) of an acute angle in a right angled triangle is defined as the ration between the side opposite the angle and the side adjacent to the angle .

       

Example 1     Find the angle  A

First 

Tan A = 3/4 = 0.75

We need to use the inverse function for tan,  tan-1, to find the angle. This function is on the same key on the calculator as the tan function (shift tan).

We use the following sequence of commands:

shift    -     tan-1   0.75     = 37º  

Try the following on your calculator to see the difference between tan and  tan-1:

angle    →    ratio                            ratio  →     angle

tan 37º     =     0.75                   tan-1 0.75    =  37º

Example 2     Find the side b

tan 37º = 4/b

tan 37º · b = 4

0.75· b = 4

b=5.3

Síne

The sine (sin) of an acute angle  in a right angled triangle is the ratio between the side opposite  the angle and the hypotenuse of the triangle.

       

Example 3     Find the angle A giving your answer to the nearest degree.

sin A = 3/5 = 0.6 gives <A = 37º

Shift sin-1  0.6  =  37º

Example 4     Find the side a.

sin 37º = a/5

a = Sin 37º · 5

a = 3

Cosine

The cosine (cos) of an acute angle in a right angled triangle is the ratio between the side adjacent to  the angle and the hypotenuse of the triangle.

      

Example 5     Use the cosine function to find the angle A giving your answer to the nearest degree.

cos A = 4/5 = 0.8 gives <A=  37º

Shift cos-1  0.8  =  37º

Example 6     Find the side b.

cos 37º = b/5

b = Cos 37º · 5

b = 4

some values for sin, cos and tan.

sin  80º = 0.98 cos 80º = 0.17 tan 80º = 5.67
sin  60º = 0.87 cos 60º = 0.5 tan 60º = 1.73
sin  30º = 0.5 cos 30º = 0.87 tan 30º = 0.58
sin  10º = 0.17 cos 10º = 0.98 tan 10º = 0.18

Practise these methods and then take trigonometry quiz 1  (sin, cos and tan).