Trigonometric Functions:
Sines, Cosines, and Tangents



Contents

  • The Right Triangle

  • The Basic Trigonometric Functions

  • Why Are These Functions Useful?

  • The Small Angle Approximation

  • Trig Functions and the Circle


    The Right Triangle

    In the course of studying astronomy (or physics) you are bound to encounter many many situations in which the right triangle will be the geometry you have to mathematically work with. Also, in situations where one encounters more complex geometries, it is always easiest to try to "break down" the problem in terms of right triangles.

    So, what is a right traingle? You may at this point want to review the sections on the definition of triangles and the types of triangles. The right triangle is basically a triangle in which one of the angles is equal to 90.

    The Basic Trigonometric Functions

    The right triangle is important to understand because it allows us to define set of so called trigonometric functions. In studying astronomy you will make occasional use of the three basic trigonometric functions: sine, cosine, and tangent. So what are these functions?

    For right triangles we give specific names to each side of the triangle (refer to Figure 1 below):

  • The longest side is called the hypotenuse and is the side that is opposite the right angle.

  • The other sides are called opposite and adjacent relative to one of the other angles. The trigonometric functions relate the lengths of the sides of the triangle to the other (i.e. not 90) enclosed angles.

    In the right triangle figure above, the side adjacent to the angle a is labeled "adj", the side opposite the angle a is labeled "opp". The hypotenuse is labeled "hyp".

    Recall that for right triangles there is a nice relationship that holds between the sides. This is called the Pythagorean Theorem and - for the above right triangle - can be written as:

    (opp)2 + (adj)2 = (hyp)2

    Furthermore, the above allows us to define the following three trigonometric functions in terms of the ratios of the sides of the triangle:

  • sin a = (opp)/(hyp) , ratio of the side opposite a to the hypotenuse

  • cos a = (adj)/(hyp) , ratio of the side adjacent a to the hypotenuse

  • tan a = (opp)/(adj) , ratio of the side opposite a to the side adjacent a

    Why Are These functions Useful?

    You might ask yourself why we bother with this ... why is it important to define the ratio of the sides in terms of more functions? Doesn't this create more things to remember? What do we gain out of defining the sine, cosine, and tangent functions?

    There are many situations in the physical universe - both in scientific discovery and in regular daily observations - where we can measure angles better than we can measure sizes or distances to objects. In such cases the trigonometric functions are essential to being able to calculate quantities which otherwise would have been extremely complicated to figure out. To give you motivation for this let us look at a simple problem of size of objects as seen from a distance.

    Angular Size, Physics Size, and Distance

    Along with giving direction and location of objects, angles can also be used to specify the size of an object. We can ask ourselves "how much angle does a particular object subtend relative to some background?"

    The angular size of an object (the angle it subtends, or appears to occupy, from our vantage point) depends on both its true physical size and its distance from us. For example, if you stand with your nose up against a building, it will occupy your entire field of view; as you back away from the building it will occupy a smaller and smaller angular size, even though the building's physical size is unchanged. Because of the relations between the three quantities (angular size, physical size, and distance), we need know only two in order to calculate the third. Suppose a tall building has an angular size of 1 (that is, from our location its height appears to span one degree of angle), and we know from a map that the building is located 10 km away. How can we determine the actual physical size (height) of the building?

     

    We imagine we are standing with our eye at the apex of a triangle, from which point the building subtends an angle of a = 1 (greatly exaggerated in the drawing). The building itself forms the opposite side of the triangle, which has an unknown height that we will call h. The distance d to the building is 10 km, corresponding to the adjacent side of the triangle. Since we want to know the opposite side, and already know the adjacent side of the triangle, we need only concern ourselves with the trigonometric tangent relationship:

    tan a = Opp/Adj

    tan a = h/d

    which can be solved as follows:

    h/d = tan 1

    h = d × tan 1

    h = 10.0 Km × tan 1 = 0.17455 km = 174.55 meters

    If you have ambiguities about the last few steps here then review the trigonometric relations and practice more problems. Also, notice the last step involving a conversion of units from km to meters.

    The Small Angle Approximation

    There are some important observations to make about the trigonometric functions in relation to actual obserable quantities and angles that were mentioned in the previous section.

    In finding the height of the building in our example above, we used the adjacent side of the triangle for the distance instead of the hypotenuse because it represented the smallest separation between us and the building. It should be apparent, however, that since we are 10 km away, the distance to the top of the building will only be very slightly farther than the distance to the base of the building; a little trigonometry shows that the hypotenuse in this case equals 10.0015 km, or less than 2 meters longer than the adjacent side of the triangle! In fact, the hypotenuse and adjacent sides of a triangle are always of similar lengths whenever we are dealing with angles that are "not very large". Thus, we can substitute one for the other whenever the angle between the two sides is small.

     

    The arclength of the circumference subtended by that small angle is only very slightly longer than the length of the corresponding straight side called "opposite". In general, then, the opposite side of a triangle and its corresponding arclength are always of nearly equal lengths whenever we are dealing with angles that are "not very large". We can substitute one for the other whenever the subtending angle is small.

    Going back to our equation for the physical height of our building from above we have:

    h = d × tan a = d × (opp/adj)

    Since the angle a is small, the opposite side is approximately equal to the "arclength" subtended by the building. Likewise, the adjacent side is approximately equal to the hypotenuse, which is in turn equivalent to the radius of the inscribed circle. Making these substitutions, the above (exact) equation can be replaced by the following, approximate, equation:

    h = d × (arclength/radius)

    But remember that the ratio (arclength/radius) is the definition of an angle expressed in radian units rather than degrees - so we have the very useful small angle approximation:

    For small angles, the physical size h of an object can be determined directly from its distance d and angular size in radians by:

    h = d ×(angular size in radians).

    The Trigonometric Functions and the Circle

    In the section on Angles and the Unit Circle we noted that in order to specify points on a circle one needs to provide a radius (a distance from a center) and an angle (which indicates direction or position along the circle).

    Making use of our defined trigonometric functions we can not specify the locations of points on a circle in a different way: we can ask how much to the right/left and how much up/down we should go to land on the specified location. This is probably ambiguous at this point, but consider the following image:

    Starting at the origin, we can get to the point on the circle by going out a distance r in the direction of the angle a. But, we can also get there by specifying a certain distance x to move to the right and certain distance y to move up. Notice that the "right" and "up" directions form right angles relative to each other - as shown above, and we get a nice right triangle with the radius as the hypotenuse. We can thus specify the location of the point on the circle in terms of the distances x and y by using our trigonometric relations:

     
    sin a = opp/hyp = y/r , ==>
    cos a = adj/hyp = x/r , ==>
    y = r × sin a
    x = r ×cos a

    We see that either way, we need two values to specify the location of a point on a circle - or of a point anywhere on the plane for that matter. The two values can be given in terms of radius and angle (r, a) or they can be given in terms of the right-left and up-down directions (x, y). The trigonometric relations are useful in relating these to ways of specifying locations of objects traveling in circular paths.

    For more information on this issue and on specifying points and locations see the section on graphing.