Trigonometry Review

 

After reading this page you should do Homework 1, Supplement 1.

 

Right Triangle Trigonometry

 

Recall that in a right triangle, little letters are used to indicate sides and capital letters are used for the angles, as in the diagram:

You should also remember the definitions of the 6 trigonometric functions, also called ratios, for each angle.  For angle A, for instance, in this picture they are as follows:

 

sin(A) = a/c

cos(A) = b/c

tan(A) = a/b

csc(A) = c/a

sec(A) = c/b

cot(A) = b/a

 

Note that the second three trigonometric functions are the reciprocals of the first three.

 

The most often taught (in American high schools, in 2004) way to remember the first three ratios is: SOH CAH TOA (sin is opposite over hypotenuse, cos is adjacent over hypotenuse, and tan is opposite over adjacent).  Then you just have to remember that the second three trigonometric functions are the reciprocals of the first three.

 

Standard position of an angle

 

An angle in standard position is an angle in the rectangular coordinate system with its initial side on the positive horizontal axis and rotated counterclockwise, as in the below diagram.

 

 

Note the meanings, indicated in diagram, of the phrases terminal side and terminal point (terminal point is not a standard term, but can be a convenient one).

 

Let  be the length of the terminal side.  (Thus, from the Pythagorean Theorem, we have ).  Then we have the very important circle definitions of the six trigonometric functions:

 

 

Signs of the trigonometric functions in various quadrants

 

Note that the value of  is always positive.  The value of y is positive in quadrants I and II and is negative in quadrants III and IV.  The value of x is positive in quadrants I and IV and is negative in quadrants II and III. 

 

These facts together with the circle definitions give us rules for the signs of all six trig functions in each of the four quadrants.  These rules are summarized in the following table.

 

 

Q I

Q II

Q III

Q IV

sin

+

+

-

-

cos

+

-

-

+

tan

+

-

+

-

csc

+

+

-

-

Sec

+

-

-

+

Cot

+

-

+

-

 

You probably have learned a convenient way to remember these sign rules: All Students Take Calculus.  The letters A, S, T, C refer to the words “all,” “sin,” “cos,” and “tan” and tell you that all trig functions are positive in Quadrant I, sin is positive in Quadrant II (and cos and tan are negative), cos is positive in Quadrant III (and sin and tan are negative), and tan is positive in Quadrant IV (and sin and cos are negative).  You have to remember also that the signs of csc, sec, and cot are the same as sin, cos, and tan, resp., because of the reciprocal identities.

 

Radian and degree measure

 

There are two common units of angle measure: degrees and radians.  An angle rotated through a full circle, so that the terminal side coincides with the initial side on the positive horizontal axis, measures 360 degrees.  This is equivalent to 2*pi radians (an asterisk means multiplication) and thus:

 

180 degrees = pi radians

 

This equivalence is usually used to convert between units of angle measure:

 

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

 

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

 

Examples:

 

a.  Convert  radians to degrees.

 

Solution:

 

b.  Convert 4.11 radians to degrees (round to the nearest tenth of a degree; note: a radian measure does not have to involve the number pi!).

 

Solution:     [done in calculator, note that your calculator has a pi key!   Use it!]

 

c.  Convert 75 degrees to radians.  Give the exact answer, in terms of pi, and a decimal approximation to 2 decimal places (nearest hundredth).

 

Solution:   [exact answer]

 

In calculator:   [decimal approximation]

 

d.  An angle of 3.89 radians will have its terminal side in what quadrant?

 

Solution: (one way to do this) Convert to degrees:

 

 

An angle of 222.9 degrees is in quadrant III, since 222.9 is between 180 and 270.

 

Special Triangles

 

You should know the following 30-60-90 right triangle and 45-45-90 right triangle.  Radian equivalents for the angles are given in the diagram.  Note: ‘sqrt’ means square root, for instance, sqrt(3) = .

 

 

 

 

These triangles enable you figure out the trigonometric functions of the special angles 30, 45, and 60 degrees. 

 

Reference Angles

 

If  is an angle in standard position, then ’s reference angle  is the positive acute angle that the terminal side of  makes with the horizontal axis.

 

Here are the formulas (in terms of degrees – but you should know them also in terms of radians) for calculating the reference angle in each quadrant, given the angle .

 

Quadrant I:

 

Quadrant II:

 

Quadrant III:

 

Quadrant IV:

 

The above formulas are used to find a reference angle when the angle itself is known.  Very often it is the case, however, that the reference angle  is known and we need to find the angle .  Thus we need to use these formulas in the following alternate forms:

 

Quadrant I:

 

Quadrant II:

 

Quadrant III:

 

Quadrant IV:

 

Using reference angles

 

The usefulness of reference angles derives primarily from the following fundamental fact: The trigonometric functions of an angle  and its reference angle  are the same except possibly for sign.  The sign is determined according to quadrant, by using the sign rules (the ASTC chart).

 

Example.  Evaluate  [no calculator]

 

Solution: step 1: find the reference angle: = [note: we’re in Quadrant IV].  Step 2: Use a special triangle to determine the trigonometric function of the reference angle: , because csc is hypotenuse over opposite.  Step 3: we know now that our answer is either  or .  To decide which, we use the sign rules (ASTC).  Our angle, , is in quadrant IV; the csc function is negative in quadrant IV and so our answer is .

 

Reference angles are also used to find angles when given the value of a trigonometric function, as in the next two examples.

 

Example.  Given , find  with .

 

Note: in the specified interval, from 0 to 360 degrees, most questions of the sort will have two answers.  (The only exceptions: some of the “boundary angles,” 0, 90, 180, and 270 degrees, but Homework 1 does not involve these).

 

Solution.  Step 1: Use a special triangle to find the positive acute angle with the given trigonometric function value, this will be the reference angle.  In this case, we see that  and so .  Step 2: Decide in what two quadrants there are answers.  In this case, there is a Quadrant I answer, say , and a Quadrant II answer, say , because these are the 2 quadrants where sin is positive (ASTC) and our given value () is positive.  For each quadrant, use the appropriate formula to find the answer:  and .

 

Example.  Given , find  with .

 

Note: this time we are to use radians (because these are the units used to specify the interval); some students like to work this type of problem in terms of degrees and then convert the answers to radians in the end.  That’s fine, but for practice working with radians, we’ll work this one entirely in radians.

 

Solution: Use a special triangle to determine the reference angle (ignore the negative sign in this step).  In this case, we see that , because cot is adjacent over opposite, and so .  Step 2:  We have answers, say  and , in Quadrants II and IV, resp., because these are the two quadrants where cot is negative.  Step 3:  and .