Pre-calc

The Law of Sines is very useful for solving triangles:
a
/sin A=b/sin B=c/sin C

a, b and c are sides. A, B and C are angles. The sides and angles correspond meaning side a faces angle A, side b faces angle B and side c faces angle C.

a^2=b^2+c^2-2bc*cos(A)
b^2=a^2+c^2-2ac*cos(B)
c^2=a^2+b^2-2ab*cos(C)

In trig, the Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles using the equations above.

Solutions for cos reflect across the x-axis
solutions for sin reflect across y-axis
solutions for tan reflect across the diagonal y=x

Solving for trigonometric equations example:
solve the equation: sin(2theta)=1/2, 0</=theta<2pi

Quadrant I
All are positive
Quadrant II
sin and cosecant are positive
cosine, tangent, secant and cotangent are negative
Quadrant III
sine, cosine, cosecant and secant are negative
tangent and cotangent are positive
Quadrant IV
sine, tangent, cosecant and cotangent are negative
cosine and secant are positive

For an angle Theta in standard position, let P=(x,y) be the point on the terminal side of Theta that is also on the circle x^2+y^2=1, the unit circle.

Standard Trig Identities
sin(Theta)=y
cos(theta)=x
tan(theta)=y/x

Reciprocal Identities
csc(theta)=1/y=1/sin(theta0
sec(theta)=1/x=1/cos(theta)
cot(theta)=x/y=1/tan(theta)

Quotient Identities
tan(theta)=y/x=sin(theta)/cos(theta)
cot(theta)=x/y=cos(theta)/sin(theta)

Quadrant I
All are positive
Quadrant II
sin and cosecant are positive
cosine, tangent, secant and cotangent are negative
Quadrant III
sine, cosine, cosecant and secant are negative
tangent and cotangent are positive
Quadrant IV
sine, tangent, cosecant and cotangent are negative
cosine and secant are positive

S.O.H C.A.H T.O.A
Fun way to remember it
Some old hippie Caught another hippie
Tripping on acid

Sine= opposite/hypotenuse or y/1

Cosine= Adjacent/hypotenuse or x/1

Tangent=opposite/adjacent or y/x

Cosecant= hypotenuse/ opposite or 1/y

Secant= hypotenuse/ adjacent or 1/x

Cotangent= adjacent/ opposite or x/y