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Trigonometric Functions of Common Angles

**The Unit Circle**** · ****Degrees
and Radians**** · ****Quadrants**** · ****Common Acute Angles and
Right Triangles**** · ****A few additional facts
about triangles**** · ****Recommended Books**

The fundamental ideas needed to understand sines and cosines can
be put into one graph, the **unit circle**:

Here we have a circle of radius r = 1 (hence "**unit circle**"),
a point (x, y) on that circle, and perpendiculars from the point to the x and y axes. The
line from the origin (0, 0) to the point (x, y) is considered a **vector** and is
usually just labeled with the endpoint (x, y). This vector forms an angle q (Greek theta) with the x-axis. The line segment from (0, 0) to (x,
y) is the longest of the three sides of the triangle and is called the **hypotenuse**.
Notice that the point on the x-axis where the perpendicular meets the axis is marked
"x = cos q" and the corresponding point on the y-axis
is marked "y = sin q". These two points represent the
meaning of the sine and cosine of the angle, normally understood as the **adjacent**
and **opposite** respectively.

Angles are given either in **degrees** (1 complete circle =
360^{o}) or **radians** (1 complete circle = 2p). For
pure mathematics, radians are preferred, although both measures of angles are in very
common use. Generally speaking, when we are interested in sine and cosine as **functions**
(of the angle q), we use radians. Radians are **pure numbers**,
the same as 0, 1, 4.3874653 and p = 3.14159…, while
degrees carry **units** (denoted with the degree sign ^{o}). Degrees require
special handling – if you use a calculator to calculate the value of trig functions
of an angle, you have to make sure you tell your calculator which units, degrees or
radians, you are using before you do the calculation.

It is reasonably easy to convert between degrees and radians. We gave the number of each unit in one complete circle above, so:

Thus, to **convert degrees to radians**, multiply by

To **convert radians to degrees**, multiply by

Example

Convert 14^{o} to radians.

Solution

Using the first of the two conversion factors above,

Two details should be pointed out here.

The first is that we did **not** write "radians"
after the answer – the lack of units is typical of pure numbers and is deliberate. If
we know the number is an angle and there is no degree sign after it, the likelihood is
that the angle has been expressed in radians. The presence of p
in the expression is another clue that the angle is expressed in radians. This is an
irrational number, and expressing it exactly requires that it be in terms of a multiple of
p.

The second detail is how we did the calculation – notice that the degree sign appears once in the numerator (we deliberately expressed 14 as the improper fraction 14/1 to give it an explicit denominator) and once in the denominator. The degrees cancel out – a valuable memory aid for those of us who can’t remember which conversion factor to use!

Example

Convert 3p/4 to degrees.

Solution

Using the second of the two conversion factors above,

Notice that angles in radians are most often expressed as
multiples of p, which helps us decide which conversion factor
to use. The important point is that we wanted our answer to be in degrees, which meant
that the degree unit needed to be in the numerator. It is possible for us to talk about
angles in radians which are **not** expressed as multiples of p,
but in practice such angles occur infrequently.

The coordinate system (also known as the x-y plane) is divided
into four regions, or **quadrants**:

Quadrant |
Allowed angles q (min, max) in degrees; radians |
Allowed values of x = cos q |
Allowed values of y = sin q |

I | (0, 90); (0, p/2) | x ³ 0 | y ³ 0 |

II | (90, 180); (p/2, p) | x £ 0 | y ³ 0 |

III | (180, 270); (p, 3p/2) | x £ 0 | y £ 0 |

IV | (270, 360); (3p/2, 2p) | x ³ 0 | y £ 0 |

It is handy to take note of the boundaries between the quadrants:

(x, y) = (1, 0): Angle with the x-axis: q = 0 Þ cos q = cos 0 = 1; sin q = sin 0 = 0.

(x, y) = (0, 1): Angle with the x-axis: q = 90^{o} =
p/2 Þ cos q = cos p/2 = 0; sin q = sin p/2 = 1.

(x, y) = (-1, 0): Angle with the x-axis: q = 180^{o}
= p Þ cos q = cos p = -1; sin q = sin p = 0.

(x, y) = (0, -1): Angle with the x-axis: q = 270^{o}
= 3p/2 Þ cos q = cos 3p/2 = 0; sin q = sin 3p/2 = -1.

These turn out to show the extreme values of both sine and cosine functions. It is
imperative to commit them to memory, along with several additional common angles. These
common angles are most often described in the first quadrant only (**acute angles**);
common angles in the rest of the quadrants are obtained from the acute angles by various
elementary operations that we will describe in a separate article.

**Common Acute
Angles and Right Triangles**** · ****The 30 ^{o}/60^{o} triangle**

**Acute** angles have measure less than 90^{o} = p/2.
There are two important **right triangles** (where one of the internal angles is
exactly 90^{o} = p/2 and the other two are acute) used
as examples to study the trig functions of the acute angles within them. The first is the
30^{o}/60^{o} triangle, the second is the 45^{o} triangle.

In keeping with the unit circle concept, we will set the
hypotenuse to have unit length, **HYP **= 1. The orientation of the triangle suggests
that the angle we are currently studying is the q = 30^{o}
= p/6 angle at the lower left corner of the triangle. If this
triangle was placed on the above graph with this corner at the origin, the base of the
triangle would lie along the x-axis and the top right corner would lie on the unit circle.
As it turns out, the remaining sides of the triangle are

We are interested in the trigonometric functions of this 30^{o}
angle. By definition,

Now, if we take this triangle and re-orient it so that the 60^{o}
angle is at the bottom left corner,

we can do a similar exercise and find the trig functions for this
angle as well. Here, **HYP** = 1 again, but the other sides of the triangle have been
reversed, so that

The angle is q = 60^{o} = p/3. Thus, the common trig functions are:

The last of the common triangles is the 45^{o} =
p/4 triangle, where both of the acute angles are the
same:

Again, **HYP** = 1, and the remaining sides are

The common trig functions are

A few additional facts about triangles

We mentioned above the definition of a right triangle - it contains a right (90^{o})
angle with the other two acute. There are several other facts about triangles that may be
helpful in later discussions.

- The sum of the three angles in a triangle is always 180
^{o}. Thus, if the triangle is a right triangle, the other two angles must sum to exactly 90^{o}. Such angles are called**complementary angles**. - If two triangles have the same three angles but different lengths of corresponding sides
(i.e., the triangles look the same, but one is bigger than the other), then the triangles
are said to be
**similar**. We are interested (for the most part) in right triangles in trigonometry and in particular those right triangles whose hypotenuses have length one (1). Studying these triangles with unit hypotenuse allows us to fit them into the unit circle. Since similar triangles have the same angles, the ratios that are used to calculate trig functions are always the same, and so we can standardize and simplify our studies considerably. In other words, studying the unit hypotenuse triangles gives the same trig information as all similar triangles. The only difference is the length of the sides, and the same factor can be used to multiply the lengths of the sides to go from a unit hypotenuse triangle to any similar triangle. - Every triangle can be broken into no more than two right triangles by drawing an appropriate perpendicular from one vertex to the opposite side. This fact can be used to analyze non-right triangles using the methods of trigonometry. More on this subject later.

Schaum's Outline of Trigonometry, 2/e

- The classic trigonometry problem book - very light on theory, plenty of problems with full solutions, more problems with answers