Triangles and Trigonometry
A triangle with angles A,B,C and opposite sides a,b,c:
What is the relationship between the angles of a triangle?
For a triangle with angles A,B,C:
A + B + C = 180 degrees
What is the relationship between the sides of a triangle (i.e. the Pythagorean Theorem)?
For a triangle with angles A,B,C and opposite sides a,b,c:
c^2 = a^2 + b^2 - 2ab*cos(C)
Notice that when C = 90, this equation becomes the Pythagorean Theorem:
c^2 = a^2 + b^2
What is the relationship between the sides and angles of a triangle? (i.e. Law of Sines)
For a triangle with angles A,B,C and opposite sides a,b,c:
sin(A)/a = sin(B)/b = sin(C)/c
What is a right triangle?
A right triangle is a triangle which has one of its angles equal
to 90 degrees (a right angle).
What is the hypontenuse of a triangle?
The hypotenuse is the side of the triangle opposite the 90 degree angle.
Only right triangles have a hypotenuse.
What are the basic trig functions?
Trigonometry involves the relationships between the sides and
angles of right-angled triangles.
H = hypotenuse (always opposite the right-angle)
O = opposite (opposite the angle theta)
A = adjacent (next to the angle theta)
sine(theta) = O cosine(theta) = A tangent(theta) = O
--- --- ---
H H A
cosecant(theta) = 1/sine(theta)
secant(theta) = 1/cosine(theta)
cotangent(theta) = 1/tangent(theta)
What are the trig identities?
Where t and u are angles:
- sin(90deg - t) = cos(t)
- cos(90deg - t) = sin(t)
- sin(-t) = -sin(t)
- cos(-t) = cos(t)
- sin(t)/cos(t) = tan(t)
- sin^2(t) + cos^2(t) = 1
- sin(2t) = 2sin(t)cos(t)
- cos(2t) = cos^2(t) - sin^2(t)
- sin(t +/- u) = sin(t)cos(u) +/- cos(t)sin(u)
- cos(t +/- u) = cos(t)cos(u) -/+ sin(t)sin(u)
What are similar triangles?
How can I find a particular angle in a triangle?
How can I find the length of a particular side in a triangle?
What is a graphical interpretation of sine and cosine?