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
= 6.43 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.
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
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
Sine Rule example
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
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.
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
PQ= 14.4 cm (to 3 sf)