Trigonometry

Triganometric ratio are used to find the angle or a side length in right-angled triangles. They are

sin x = opp/hyp (SoH)

cos x = adj /hyp (CaH)

tan x = opp/adj (ToA)

 hyp= 10cm angle = 40 . To find the opposite

sin x= opp / hyp

sin 40= a/10

a=  10 X sin 40

= 6.42787

= 6.43 cm (to 3 sig fig)

a² + b² = c²  <-- the main formula

Always label the sides with a, b and c. A and B are the two shorter sides. C is always the hypotenuse.

Example

A triangle with a=4cm c= 11cm. Find x cm of b.

4² + x² = 11²

x² = 11² - 4²    = 121 - 16  = 105

x = 10.2 cm

The sine rule applies to any triangle it doesn't have to be a right-angled triangle.

You label the angle of the triangle with capital letters and the sides with lower case letters. Each side has the same letter as it's OPPOSITE angle.

a/Sin A = b/Sin B= c/Sin C  <- Use this to find a missing side

Sin A/ a = SinB/b= SinC/c <- Use this to find a missing angle

Length a= 8cm b=? c=14cm

Angle a= x b=? c= 106

Sin A/a = Sin C/ c

Sin X/8 = Sin 106/14

Sin x = 8X Sin 106/ 14

= 0.5492923977 (to the -1)

x= 33.3 (to 3 sf)

The cosine rule applies to any triangle. You don't need a right angle.

a² = b² + c² - 2bc cos A <- this version is used to find the missing side

cos A = b²  + c² -  a² / 2bc <- is used to find the missing angle.

Example;

pq = a   b=8cm  c=15cm   angle PRQ = 70

a² = b² + c² - 2bc cos A

PQ² = 8² + 15² - 2 X 8 X 15 X cos 70

=206.9151..

PQ= 14.4 cm (to 3 sf)