Patrick Teaches Physics

What are "cos" and "sin"?  They're buttons on a calculator -- right?  You enter an angle, push the button, and some number appears.  Somehow -- if you remember your trig rules (SOH, CAH, TOA) -- you can use this number get the side lengths of triangles. I think the above pretty much sums up the typical student's understanding, and it's adequate to complete most of the tasks a physics student has to with trig functions.  But it doesn't have to be so mysterious, and it can be very helpful if it isn't.  So in this post I'll explain what these functions are and how they are related. Consider trying to specify a point on the unit circle: We could use the Cartesian coordinates x and y, or we could use the angle θ. Both have their merits, and in practice we might want to go back and forth between them.  So, given an angle θ, what are x and y?  We call the functions that answer that question sine and cosine (usually written as sin and cos for sho...

You grind your way through a problem, get an answer, and want to know if it is correct.  How can you tell? In the previous post I talked about changing the angle in a problem (the inclined plane) to see how that affected other angles.  It can help you see how the angles relate to each other.  In this post I will use the same technique, but this time to test if my answer could be wrong. Let's return to the inclined plane.  Suppose a block of mass m = 2.5 kg slides down a frictionless plane at a slope of θ = 35 degrees to the horizontal.  You need to determine the block's acceleration.    You calculate an acceleration of a = 8.0 m/s/s down the slope.  Your friend, attempting the same problem, gets an answer of a = 5.6 m/s/s.  Who is right? After both checking that your calculators are set to degrees (not radians), you look at each other's work.  You have a = g cos θ while your friend has a = g sin θ where g is the usual accelerat...

When I first encountered the radian it seemed that the world had just become unnecessarily more complicated.  What was wrong with plain old degrees , and how did I know when to use which?  I found out it really helps to learn what a radian actually is . Consider a circle of radius r , and imagine wanting to know the length of an arc, s , subtended by an angle θ. Suppose the angle is such that s = r.  That angle actually defines the unit radian, so in that case θ = 1 radian.  Obviously, the arc length is proportional to the angle, so we get the general formula  s = rθ so long as we measure the angle in radians.  That's what makes the radian so useful: it gives a direct relationship between angular distance and length.  One place that has obvious applications is in rotational kinematics, because it allows us to easily relate angular velocity to tangential velocity. But how big is a radian in degrees?  The circumference of a circle is the arc len...