• 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)

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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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.

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!!

• 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.