Basics of Fluid Mechanics, 2014a

A.4. TRIGONOMETRY 595
A.4 Trigonometry
These trigonometrical identities were set up by Keone Hon with slight modification
1. sin(α + β) = sin α cos β + sin β cos α
2. sin(α − β) = sin α cos β − sin β cos α
3. cos(α + β) = cos α cos β − sin α sin β
4. cos(α − β) = cos α cos β + sin α sin β
5. tan(α + β) =
tan α + tan β
1 − tan α tan β
tan α − tan β
6. tan(α − β) =
1 + tan α tan β
1. sin 2α = 2 sin α cos α
2. cos 2α = cos 2 x − sin 2 x = 2 cos 2 x − 1=1− 2 sin 2 x
3. tan 2α = 2 tan α
1 − tan 2 α
4. sin α √
1 − cos α
2 = ± (determine whether it is + or - by finding the quadrant
2
that α lies in)
2
5. cos α √
1 + cos α
2 = ± (same as above)
2
6. tan α 2 = 1 − cos α = sin α
sin α 1 + cos α
for formulas 3-6, consider the triangle with
sides of length a, b, and c, and opposite angles α,
β, and γ, respectively
1. sin 2 α = 1 − 2 cos(2α)
2
2. cos 2 α = 1 + 2 cos(2α)
2
3.
sin α
a
= sin β
b
= sin γ
c
(Law of Sines)
4. c 2 = a 2 + b 2 − 2 abcos γ (Law of Cosines)
5. Area of triangle = 1 2 absin γ
β
a
c
γ
α
b
Fig. -A.7. The tringle angles sides.
6. Area of triangle = √ s(s − a)(s − b)(s − c),
where s = a + b + c (Heron’s Formula)
2