15 ## ANALYTIC TRIGONOMETRY## THE UNIT CIRCLEANALYTIC TRIGONOMETRY is an extension of right triangle trigonometry. It takes place on the Let a radius of length We will take our cue from the first quadrant. In that quadrant, a radius
According to the Pythagorean theorem, In this way we extend the meaning of the trigonometric functions to angles that terminate in But before we give an example, consider this question: Will a function of θ depend on the length of To see the answer, pass your mouse over the colored area.
No, it will not. The functions are defined as the Say that AB, AC are two different radii. But triangles ABD, ACE are similar. Proportionally, DB : BA = EC : CA sin θ -- opposite over hypotenuse -- does not depend on the The trigonometric functions in fact depend only on the angle θ -- and it is for that reason we say that they are functions of θ. Example 1. A straight line inserted at the origin terminates at the point (3, 2) as it sweeps out an angle θ in standard position. Evaluate all six functions of θ.
Problem 1. A straight line from the origin sweeps out an angle θ, and it terminates at the point (3, −4). Evaluate the six functions of θ.
Problem 2. The signs in each quadrant. a) The algebraic sign of sin θ will always be the sign of which a) Therefore, in which quadrants will sin θ -- a) In which quadrants will sin θ be negative? III and IV. b) The algebraic sign of cos θ will always be the sign of which positive. a) Therefore, in which quadrants will cos θ -- a) In which quadrants will cos θ be negative? II and III. c) In which quadrants will the algebraic sign of tan θ ( I and III. d) In which quadrants will the algebraic sign of tan θ be negative? II and IV. e) csc θ will have the same sign as which other function? sin θ, because they are reciprocals. f) sec θ will have the same sign as which other function? cos g) cot θ will have the same sign as which other function? tan Quadrantal angles A quadrantal angle is an angle that terminates Problem 3. a) What are the quadrantal angles in degrees? 0°, 90°, 180°, 270°; and angles coterminal with them. b) What are the quadrantal angles in radians?
c) When an angle terminates on the d) When an angle terminates on the Now, it is a fact of arithmetic that there is no
Therefore, wherever a trigonometric function has a denominator -- For example,
Wherever tan θ has no value at θ = and at , or at −, which is coterminal. Those values of θ will be singularities of tan θ. (Topic 18 of Precalculus.) In calculus, the student will encounter the expression, "The limit of as As explained in Topic 4 of calculus, the word infinity along with its symbol ∞, is not a number and it is not a place. When we say that a function becomes infinite, we simply mean that there is no limit to its values. The function will have values greater than any number that we name. Note that we say, "As Problem 4. For which quadrantal angles do the following functions have no value?
the b) sec θ
c) sin θ
sin θ has no value. The unit circle The trigonometric functions are functions only of the angle θ. Therefore we may choose any radius we please, and the simplest is a circle of radius 1, the unit circle. On the unit circle the functions take a particularly simple form. For example,
The value of sin θ The value of cos θ is the The following shows the coördinates of the endpoint on the unit circle of every quadrantal angle: If a function has a value at a quadrantal angle, Consider sin θ at each quadrantal angle. We just saw that the value of sin θ sin θ = Therefore at each quadrantal angle, the value of sin θ -- of
To evaluate a function at a quadrantal angle, the student should sketch a unit circle. Problem 5. Evaluate the following. No tables. a) cos 0° cos 0° = 1. cos θ is equal to the b) cos 90° = 0 c) cos 180° = −1 d) cos 270° = 0 e) tan 0° tan 0° = 0. tan θ is equal to f) tan 90° 1/0 has no value. g) tan 180° = 0 h) tan 270° has no value. Problem 6. Evaluate the following -- if it has a value. No tables.
Problem 7. Explain why we can write the following, where cos
(−1) See the unit circle. Radian measure Since in any circle the same ratio of arc to radius determines a unique central angle, then for theoretical work we often use the unit circle. The radian measure is the ratio of arc length
We can identify radian measure, then, as the length It is here that the term trigonometric "function" has its full meaning. For corresponding to each real number Moreover, when we draw the graph of Because radian measure can be identified as an arc, the inverse trigonometric functions have their names. "arcsin" is the arc -- the radian measure -- whose sine is a certain number.
One of the main theorems in calculus concerns the ratio
for very small values of In the unit circle, side AB opposite angle AOB
We can see that when the point A on the circumference is very close to C -- that is, when the central angle AOC is extemely small -- then the side AB will be virtually indistinguishable from the arc length AC, which is the radian measure. That is,
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