LEAVE A COMMENT FOR US

Home > Trigonometry > Exact Values of Trigonometric Ratios

Exact Values of Trigonometric Ratios

exact values of trigonometric ratios-1To 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.

exact values of trigonometric ratios-2AD2  =  AB2  –  BD2

=  22  –  12  =  4 – 1  =  3

AD = sqrt 3 unit.

Using these values, we can get

sin 60° =  {opposite}/{hypotenuse}  =  AD/AB  =  sqrt {3}/2; and

sin 30°  =  {BD/AB}~~=~~{1/2}

cos 60° =  {adjacent}/{hypotenuse}  =  BD/AB  =  1/2; and

cos 30°  =  {AD/AB}~~=~~sqrt {3}/2

tan 60° =  {opposite}/{adjacent}  =  AD/BD  =  sqrt {3}; and

tan 30°  =  {BD/AD}~~=~~1/{sqrt 3}

 

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

exact values of trigonometric ratios-3AC  =  sqrt {AB^2~+~BC^2}~~=~~sqrt{1^2~+~1^2}~~=~~sqrt 2

AC = sqrt 2 units.

sin 45°  =  AB/AC~~=~~1/{sqrt 2};

cos 45°  =  BC/AC~~=~~1/{sqrt 2}; and

tan 45°  =  AB/BC~~=~~1/1  =  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°  =  {1/2}~+~{1/2}  = 1

 

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

sin 60° + sin 30° =  {sqrt 3}/2~+~1/2~~=~~{sqrt 3~+~1}/2

 

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

2sin 30°cos 30° =  2~*~{1/2}~*~ sqrt 3/2~~=~~ sqrt 3/2  =  sin 60°

 

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

sin² 30° + cos² 30° =  (1/2)^2~+~(sqrt {3}/2)^2

= {1/4}~+~{3/4}  =  1

 

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

1 – 2sin² 30°  =  1 – 2 x (1/2)^2

=  1 – 1/2~~=~~1/2  =  cos 60°

 

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

2cos² 30° – 1  =  2 x (sqrt {3}/2)^2 – 1

=  2 x 3/4  – 1

=  3/2 – 1

=  1/2  =  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