Problems on Trigonometry

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Section A: Angles (units and conversions)

1. Convert the following angles to radians.

a. 30º = d. 18º = g. 352º =
b. 60º = e. 97º = h. 360º =
c. 15º = f. 163º = i. 270º =


2. Convert the following angles to degrees (and to arcseconds/arcminutes if you wish to practice more!).

a. p radians = d. 6.19 radians = g. 64 radians =
b. 2p radians = e. 0.52 radians = h. p/6 radians =
c. 3.0 radians = f. 0.0016 radians = i. 4p/3 radians =


3. Convert the following angles to radians (you might want to recall the definition of arcminutes and arcseconds):

4. Conver the following from radians to degree, arcminute, arecsecond notation:


Section B: Trigonometric Functions and Right Triangles

1. Evaluate the following quantities. This is for you to practice taking the sines and cosines and tangents of angles that are given to you in radians or in degrees.

a. sin (p/2) = ? d. cos (p/2) = ? g. tan (p) =?
b. sin (42º) = ? e. cos (42º) = ? h. tan (45º)
c. sin (162º) = ? f. cos (162º) = ?  


2. Suppose that instead of being given the angle you encounter a situation - for instance in setting up a problem or solving for a quantitiy in a given situation - in which you know what the sine or cosine of some angle is equal to, but you don't know the angle! In these cases you take the inverse of the function to get the angle. Practice doing the following to see how this is done. On your calculator you simply do the "inverse" (often called <shift> or <2nd>). Sometimes - if the number is a nice number (such as b. or f. below) you ask yourself: "what angle would I have to take the sine/cosine of in order to get 0.0 or 1.0?".

a. sin (x) = 0.2, x = ? d. cos (x) = 1.0, x = ?
b. sin (x) = 0.0, x = ? e. cos (x) = 0.34, x = ?
c. sin (q) = 1.9, q = ? f. cos (q) = -1.0, q = ?


3. A right triangle has one leg equal to 3.0 cm. The angle opposite to this leg is measured to be 30º. Calculate the length of the other leg and the length of the hypotenuse. Refer to the figure below.


Section C: Small Angle Approximation, Arc Length ... and more

1. For which of the following situations drawn below is the small angle approximation - namely that sin(q)=q - valid?


2. A hill is at a distance of 10.0 km from you. You stand up and measure the angular size that it spans and you notice that is is about 2º. From this information, will you be able to determine the height of the hill? If so, what is the height? If not, explain why not?


Section D: Solving

1. Given "a" = (7/8)π radians, "b"= 70 degrees, and "d"= 80 degrees: Express angle "c" in:

a) degrees
b) radians
c) arcminutes and arcseconds

2. find angle alpha in the following diagram:

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