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A trigonometric function is a function that relates a measure of an angle of a right
triangle to a ratio of the triangle’s sides.
The basic trigonometric functions are cosine and sine. They are called “trigonometric”
because they relate measures of angles to measurements of triangles. Given a right
we define Note, the values of sine and cosine do not depend on the scale
of the triangle. Being very explicit, if we scale a triangle by a scale factor
At this point we could simply assume that whenever we draw a triangle for
computing sine and cosine, that the hypotenuse will be . We can do this because we
are simply scaling the triangle, and as we see above, this makes absolutely no
difference when computing sine and cosine. Hence, when the hypotenuse is , we find
that a convenient way to think about sine and cosine is via the unit circle:
If we consider all possible combinations of ratios of
adjacent, opposite, hypotenuse,
(allowing the adjacent and opposite to be negative, as on the unit circle) we obtain all
of the trigonometric functions.
The trigonometric functions are: where the domain of sine and cosine is all real
numbers, and the other trigonometric functions are defined precisely when their
denominators are nonzero.
Which of the following expressions are equal to ?
Note, , and since they differ when .
Not all angles come from triangles.
Given a right triangle like
the angle cannot exceed radians. That means to talk about trigonometric functions
for other angles, we need to be able to describe the trigonometric functions a little
more generally. To do this, we use the unit circle from the previous section. Given an
angle , we construct the angle with initial side along the positive -axis and vertex at
As the angle grows larger and larger, the terminal side of that angle spins around the
circle. The trigonometric functions of the angle are defined in terms of the terminal
Suppose is the point at which the terminal side of the angle with measure
intersects the unit circle. The the trigonometric functions are defined to be provided
these values exist.
From the picture, you see that this agrees with what you know about trigonometry
for triangles, but it allows us to extend the definition of sine and cosine to all real
numbers, instead of only the interval
As a reminder, we include the graphs here.
The power of the Pythagorean Theorem
The Pythagorean Theorem is probably the most famous theorem in all of
Pythagorean Theorem Given a right triangle:
We have that:
The Pythagorean Theorem gives several key trigonometric identities.
Pythagorean Identities The following hold:
From the unit circle we can see
via the Pythagorean Theorem that If we divide this expression by we obtain and if
we divide by we obtain
There several other trigonometric identities that appear on occasion.
If we plug into the angle addition formulas, we find the double-angle identities.
Solving the bottom two formulas for and gives the half-angle identities.
Frequently we are in the situation of having to determine precisely which angles
satisfy a particular equation. The most basic example is probably like this one.
We’ll start by finding the reference angle, , the acute angle
between the terminal side of and the -axis. The reference angle satisfies .
From the picture we see . In one period , there are two angles that have reference
angle and have negative sine value. One is in quadrant 3, and one in quadrant 4.
That means the solutions in the interval are and .
To find all solutions, we have to add all multiples of to these. The solutions are then
Solve the equation:
None of the above
Let’s try one a bit more complicated.
Solve the equation:
We’ll start by simplifying a
Notice that this equation is quadratic in . We can factor it like we try to do to solve
any other quadratic equation: On the interval , has only one solution, . For , we see
that the reference angle . Since cosine is positive in quadrants 1 and 4, we find
solutions and .
All solutions are:
Limits involving trigonometric functions
Back when we introduced continuity we mentioned that each trigonometric function
is continuous on its domain.
Compute the limit:
The multiplicative limit law allows us to split this into The
function is continuous everywhere, so . Since is in the domain of , we have . Putting
these together we find
Compute the limit:
Recall that , so that Since sine and cosine are continuous, and .
That is, is of the form .
The numerator is negative for near . From the graph of , we know that the
denominator is negative and approaching as . That means
Compute the limit:
We’ll end with a couple very involved limits where the Squeeze Theorem makes a
To compute this limit, use the Squeeze Theorem. First note that we only
need to examine and for the present time, we’ll assume that is positive. Consider
the diagrams below:
From our diagrams above we see that and computing these areas we find
Multiplying through by , and recalling that we obtain Dividing through by and
taking the reciprocals (reversing the inequalities), we find Note, and , so these
inequalities hold for all . Additionally, we know and so we conclude by the Squeeze
When solving a problem with the Squeeze Theorem, one must write a sort of
mathematical poem. You have to tell your friendly reader exactly which functions
you are using to “squeeze-out” your limit.
Let’s graph this function to see what’s going on:
The function has two factors:
Hence we have that when and we see and so by the Squeeze theorem, In a similar
fashion, when , and so and again by the Squeeze Theorem . Hence we see that