Devise; Don't Memorize.

One of the common pitfalls in using right-triangle trigonometry for physics problems is incorrectly assuming that the cosine function always goes with the "x component" of a vector, and the sine function always goes with the "y component".

Which axis is associated with which trig function depends on how a particular angle is defined: that is, which of the two cartesian axes is the reference.

In the sketch on the left, the vector's angle with the x axis is given so its x component is A_x=Acos(26^o). In the sketch on the right, the vector's angle with the y axis is given so its x component is A_x=Asin(64^o). Since the sine of an angle is equal to the cosine of its compliment, there is no real difference between the two calculations. The way most physics textbooks start you thinking about components is with the x axis as reference, so you may find yourself memorizing the equations

A_x=Acosq A_y=Asinq.

There are some important equations in physics that involve the sine and cosine functions. For example:

work
W = Fdcosq

torque
t = rFsinq

Snell's Law
n₁sinq₁ = n₂sinq₂.

In each case, the "q" is measured with respect to a particular reference line or vector that can be horizontal, vertical, or neither. It is important to treat each problem without assumptions about the reference, and devise the appropriate trigonometric equation from the particular geometry rather than resort to a formula memorized from the textbook.

Here is a diagram taken from a physics test on torque equilibrium analysis. The stick has a hinge at the upper end and a weight tied to the lower end, and a string (providing tension force T) tied 80 centimeters from the hinge holds up the system. The question: what is the angle that should be plugged into the equation for torque? In the torque equation t=rFsinq, the angle q is the angle between the force (the string) and the lever (the stick), which is neither of the angles shown here (q=60^o).

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