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# Exact Values of Trigonometric Ratios

To find the exact values of trigonometric ratios, consider an equilateral triangle ABC of sides 2 units.

Draw AD perpendicular to BC. AD is a perpendicular bisector (as AB = AC), and forms two isosceles triangles, with the altitude bisecting the third side in isosceles triangle. So BD = DC = 1 unit.

Now consider a right-angled triangle ABD. AD is perpendicular to DB, AB = 2 units and BD = 1 unit.

=  22  –  12  =  4 – 1  =  3

Using these values, we can get

sin 60° =    =    =  ; and

sin 30°  =

cos 60° =    =    =  ; and

cos 30°  =

tan 60° =    =    =  ; and

tan 30°  =

Now consider a right-angled isosceles triangle ABC with 1 unit of equal sides.

AC  =

AC = units.

sin 45°  =  ;

cos 45°  =  ; and

tan 45°  =    =  1

Now let us look at some examples of calculating exact values of trigonometric ratios:

Example 1: Find the exact value of sin 30° + cos 60°

sin 30° + cos 60°  =    = 1

Example 2: Find the exact value of sin 60° + sin 30°

sin 60° + sin 30° =

Example 3: Find the exact value of 2sin 30°cos 30°

2sin 30°cos 30° =    =  sin 60°

Example 4: Find the exact value of sin² 30° + cos² 30°

sin² 30° + cos² 30° =

=  =  1

Example 5: Find the exact value of 1 – 2sin² 30°

1 – 2sin² 30°  =  1 – 2 x

=  1 –  =  cos 60°

Example 6: Find the exact value of 2cos² 30° – 1

2cos² 30° – 1  =  2 x – 1

=  2 x  – 1

=   – 1

=    =  cos 60°

Note  Examples 3, 4, 5, 6 can be given as trigonometric identities as stated below:

• 2 sin θ cos θ  =  sin 2θ
• sin² θ  +  cos² θ  = 1
• 1 – 2sin² θ  =  cos 2θ
• 2 cos² θ – 1  =  cos 2θ