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. (Theorem 15) Proportionally, DB : BA = EC : CA sin θ -- opposite over hypotenuse -- does not depend of 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
coterminal. Those values of θ will be singularities of tan θ. (Topic 18 of Precalculus.) Problem 4. For which quadrantal angles do the following functions not exist?
the b) sec θ
c) sin θ
sin θ does not exist. 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 With regard to quadrantal angles, the unit circle illustrates the following: If a function exists 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 does not exist. g) tan 180° = 0 h) tan 270° does not exist. Problem 6. Evaluate the following -- if it exists. No tables.
Problem 7. Explain why we can write the following, where cos
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