Chapter 4, Section 3

The trigonometric convention:

Let’s talk about angles and rotations – both positive and negative. We’ll initially confine ourselves to rays that emanate from the origin and intersect the unit circle:

Note that the equation that makes the unit circle is

(not a function!)

Our initial ray is always along the positive x axis. Our terminal ray can move to any quadrant.

We define q in radian measure

Where is q = ?

We define the point pairs on the unit circle to be (cos q, sin q). We have, then, that

.

In fact we define each of the six trigonometric functions in terms of x and y coordinates on the unit circle:

sin q = y csc q =

cos q = x sec q =

tan q = y/x cot q =

Note that the values for the trigonometric functions can be either negative, positive, zero or undefined.

Note now how I specify quadrants in radian measure:

Q1 0 < q < Q3

Q2 Q4

Suppose

fill in either < or > in the following statement:

cos a + sin b ______0.

This is a typical kind of problem.

Suppose you have a point P on the unit circle and you don’t know q but you do know one point coordinate:

for example x = - 4/5 . This puts P in Q2 or Q3.

Suppose further that I tell you .

You can tell me all the other trig function values for this q.

Let’s see how:

In Quadrant 1 the following are very important:

Know this by heart!

angle in deg / 0 / 30 / 45 / 60 / 90
angle in rad
sine
cosine
tangent

On the boundaries of the quadrants: Quadrantal Angles

for example:

list the 6 trig function values:

Looking at angles that reference negative and positive

Write down all the angles with their cosines and sines:

Locate and evaluate:

+ =

And those referencing : negative and positive

Write down all the angles with their cosines and sines:

Locate and evaluate:

+ =

And those referencing : negative and positive

Write down all the angles with their cosines and sines:

Locate and evaluate:

+ =

Evaluate

If sin t = , compute cot t.

If cos x = -3/5 and , compute the remaining 5 trig functions.


And just for good measure:

Area of a triangle:

Thetas:

30°

45°

60°

90°

120°

150°

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