## Trigonometry

### BASICS:(top)

Trigonometry is often referred to as the study of triangles, as, indeed, it is. Yet trigonometry also concerns itself with the relationships between angles in general.

Measuring angles is the most fundamental skill of trigonometry. There are two units that are commonly used to measure angles: degrees and radians. For both units, angle measurement starts at the positive x-axis, the initial side, and is measured counterclockwise for a positive angle and clockwise for a negative angle.

 positive angle negative angle

Degrees

Degrees are the unit of angle measurement more familiar to most students. Angles are measured on a circle, which has 360˚. Each quarter of the circle, therefore, is 90˚. The right-hand side of the x-axis is designated as the 0˚ mark.

While radians are often less familiar to students, they are in fact often much more useful than degree measurements. One radian is equal to the angle subtended by the center of a circle of an arc that is equal in length to the radius of the circle, as shown at right. This corresponds to approximately 57.3˚. A full circle as 2π radians, meaning the quarters of the circle are at π/2, π, and 3π/4, as shown below.

EXAMPLE 1:
Problem: Mark 330˚ and 3π/4 radians on coordinate systems.

 Solution: 330˚ is 60˚ more than 270˚.   3π/4 is π/4 more than π/2.

Note that to convert from degrees to radians, one should multiply by π/180:

To convert from radians to degrees, mutiply by 180/π.

EXAMPLE 2:
Solution: To convert from radians to degrees, multiply by 180/π:

To convert from degrees to radians, multiply by π/180:
.

### TRIG FUNCTIONS:(top)

Basic Trig Functions

There are 3 basic trigonometric functions: sine (sin), cosine (cos), and tangent (tan). These functions determine certain relationships between the angles in a right triangle and its sides. Note that in a right triangle, the legs are the two sides that form the right angle, while the hypotenuse is the side opposite the right angle.

The sine, cosine, and tangent of an angle are defined as follows:
; ; .

EXAMPLE 3:
Problem: Evaluate the sine, cosine, and tangent of the angles and ø in the triangle at left.
Solution: First, use the Pythagorean theorem to find that the hypotenuse has a length of 5. (Since 32+42=c2, c2=25, so c=5.) Then use the definitions above to find:

 sin= 4/5 sinø= 3/5 cos= 3/5 cosø= 4/5 tan= 4/3 tanø= 3/4

Reciprocals

There are also 3 reciprocal trigonometric functions: cosecant (csc), secant (sec), and cotangent (cot). These functions are the reciprocals of sine, cosine, and tangent, respectively:

;
;
.

Inverses

Finally, each of the trigonometric functions we have covered so far has a corresponding inverse function.

### SPECIAL TRIANGLES AND ANGLES:(top)

It is important to realize that we cannot, in general, easily compute values of trigonometric functions. There is often no easy, calculator-free way of evalutating something like sin(37˚). We do, however, have some ways of computing a number of special angles, such as 0˚, 30˚, 45˚, 60˚, and 90˚ using some special triangles; we will then develop ways of evaluating some other angles (part 1, part 2) from the ones we do know.

The 45˚- 45˚- 90˚ Triangle

The first special triangle we'll look at has angles of 45˚, 45˚, and 90˚. Since the two acute angles are equal, the two legs of the triangle will also be equal. Let's designate their length as a. Since this is a right triangle and we know the lengths of its legs, we can find the length of its hypotenuse—which we'll designate as c—using the Pythagorean theorem:

Now, we can use the triangle to find the sine, cosine, and tangent of 45˚:

And of course, we can find the cosecant, secant, and cotangent of 45˚ by taking the reciprocals of sine, cosine, and tangent, respectively.

The 30˚- 60˚- 90˚ Triangle

The second special triangle we're concerned with has angles of 30˚, 60˚, and 90˚ and will allow us to determine the values of basic trigonometric functions for =30˚ and =60˚. This triangle is shown in the diagram at right. At first glance, we have no obvious way of determining the lengths of any of its sides. If we note that this triangle is half of the larger equilateral triangle shown below, however, things get much easier.

We're going to let each side of the equilateral triangle equal 2a. Since the base of our original triangle is half the length of the base of the equilateral triangle, it has length a. We can then use the Pythagoren theorem to determine the height of our original triangle, which we'll call b:

Now that we have the complete triangle, we can use the definitions of sine, cosine, and tangent to calculate their values at 30˚ and 60˚.

And of course, we can find the cosecant, secant, and cotangent of 30˚ and 60˚ by taking the inverses of each angle's sine, cosine, and tangent, respectively.

### SPECIAL ANGLES ON THE CIRCLE:(top)

So far, we have looked only at acute angles within a triangle. We can use what we have learned there to calculate trigonometric functions for other values around the circle as well, such as 135˚ and 11π/6. This requires introducing the concept of the reference angle. For any given angle, its reference angle is the smallest angle made with the positive axis.

A reference angle is the internal angle of the triangle formed by the original angle. Consequently, the values of trigonometric functions evaluated at the reference are identical to those of the original angle, with the possible exception of the sign.

 sine/cosecant cosine/secant tangent/cotangent

### GRAPHING TRIG FUNCTIONS:(top)

f(x)=A*trig(Bx+C)+D

A=amplitude; B=affects the period; C=phase shift; D=moves up and down

EXAMPLE 4:
Problem:

Solution:

### IDENTITIES:(top)

Being able to change a trigonometric expression into another, equivalent form is a very important skill. It is useful for simplifying and evaluating expressions for differentiation or integration as well as for solving trig equations. One can manipulate trig expressions using the identities below. Click for explanations.

Reciprocals
a); ;
b); ;
c); ;
Explanation
The first two identities in each row are true by definition (see above), and the last in each row follows immediately from those definitions.
Quotients
a)     b)
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Co-functions
a)     b)     c)
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Even/Odd
a)     b)     c)
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Pythagoreans
a)
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b)

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c)
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Sum/Difference
 a) Text Explanation b) Text Explanation c) d) e) e)

Double Angle
a)
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b)
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c)
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Half Angle
a)
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b)
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c)
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Let's look at some of the uses for trigonometric identities: evaluating trigonometric functions, simplifying trig expressions, and solving trig equations.

EXAMPLE ?:
Problem: Simplify the trigonometric expression .
Solution: Use the identites given above as follows:

 by FOIL in the numerator by the Pythagorean identity by the quotient identity by canceling out the sin2x terms by the definition of secant

EXAMPLE ??:
Problem: Evaluate sin(75˚) without a calculator.
Solution: Notice that 75˚ = 30˚ + 45˚. We can then use the addition formula for sine, , to evaluate sin(75˚) as sin(30˚+45˚):

 sin(30˚+45˚) =sin(30˚)cos(45˚)+cos(30˚)sin(45˚) = * + * = + =