STUDY GUIDE: Trigonometry
INTRODUCTION
HERE'S a different approach
  • Despite it's rather intimidating name, trigonometry is simply the study of the relationships between the angles and the sides of right triangles.
  • There are three main functions in trig: sine (sin), cosine (cos) and tangent (tan).
  • There are three insignificant functions in trig: secant, cosecant and cotangent. These are kind of redundant functions and simply represent the (1/cosine, 1/sine and 1/tangent) values of the main function listed above. Forget about them... well at least for now. We will *rarely* if ever use them, and you'll figure them out when and if....
  • Each of the main function in trig relates the ratio of two sides of a right triangle to the angle between those sides.
    • The side of a right triangle opposite the 90 degree angle is the hypotenuse
    • Because one angle of a right triangle is always 90 degrees, the other two angles of a right triangle must measure more than 0 degrees and less than 90 degrees
    • In calculations using trig, we almost always use one of the other two angles (not the right angle)
 
THE MAIN FUNCTIONS: SIN, COS & TAN
 

triangle

 

Consider the right triangle above

  • note the right angle (90 degrees) between sides a and b
  • note sides of length a and length b, with hypotenuse of length c
  • note angle "Θ" (It's customary in science & math to use the Greek letter "theta" (Θ) to represent angles)
    • sin(Θ) is defined as the ratio of side opposite angle Θ (side a) to the hypotenuse (side c): therefore sin(Θ) = a/c
    • cos(Θ) is defined as the ratio of the adjacent angle Θ (side b) to the hypotenuse (side c); therefore cos(Θ) = b/c
    • tan(Θ) is defined as the ratio of the side opposite angle Θ (side a) to the side adjacent to angle Θ (side b). Therefore tan(Θ) = a/b

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    sohcahtoa (pronounced sew-kah-toe-ah) can be helpful. It stands for Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse and Tangent = Opposite over Adjacent

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WHY WE CARE

If we know the angle of a right triangle and just one of the other sides we can find the lengths of the other sides.

Check this out!

Let's say that angle Θ above is actually 30 degrees. And we know that the length of side a = 10 meters.

Using the definition above, we know that sin(30 degrees) = length of the side of the triangle opposite angle Θ divided by the length of the hypotenuse.

Writing that using math terms, we get:

sin(Θ) = a/c.

If we plug in the angle we know (30 degrees) and the length of side a, we get:

sine(30 degrees) = 10 meters/(length of hypotenuse)

plugging in sine(30) into our calculator:

we get a value of .5

Substituting above:

.5 = 10 meters/(length of hypotenuse)

substituting the letter c for the (length of the hypotenuse) we get

.5 = 10 meters/c

and solving for c we get

.5c = 10 meters

c= 10 meters/.5

c=20 meters

In physics, we frequently talk about the magnitude of a vector. That magnitude is frequently the hypotenuse of a triangle and often symbolized as R. As such, we can often calculate the x component of that vector by Rcos(Θ) while the y component of the vector is calculated by Rsin(Θ). Where Θ is the angle between the vector and the x axis.

INVERSE FUNCTIONS

Once again the terms are somewhat more intimidating than the functions. Inverse trig functions simply relate known sides to angles.

In physics we sometimes know the sides of a triangle but the angle between them is unknown.

For example. In the example above. If we knew the side opposite angle Θ was 10, and we knew the hypotenuse was 20 then we would write the information we know like this:

sin(Θ) = 10/20

sin(Θ) = .5

Our thinking would be:

"There is an angle Θ where the ratio of the side opposite angle Θ to the hypotenuse is equal to .5"

using the inv button on our calculator together with the sin button we type in:

(.5)(inv)(sin)

AND VOILA!!! Out pops 30 degrees!!

TRIG IN PHYSICS

We almost *always* draw figures when we solve problems in physics. Throughout this term you will find references to angles and triangles and it is critical that you place those triangles in the correct quadrant of the coordinate system.
 
coords
Consider the cartesian coordinate plane system shown above with quadrants labeled as I, II, III and IV.
 
  • We always label angles in trig begriming on the positive x axis
  • Positive angles are measured counter clockwise from the positive x axis
  • FOR EXAMPLE:
    • Moving from a point on the positive x axis to a point on the positive y axis involves a movement of 90 degree
    • Continuing to move to a point on the negative x axis involves another 90 degree turn resulting in a 180 degree movement
    • Continuing to move to a point on the negative y axis involves another 90 degree turn resulting in a total of 270 degrees
    • A final 90 degree turn brings us back to where we started from, for a total of 360 degrees.
X values are always positive in quadrants I and IV while Y values are always positive in quadrants I and II.
 
Triangles drawn with x components in those quadrants will behave exactly the same way.