Download TRIGONOMETRY REVIEW Calculus Appendix B ANGLES Radian

TRIGONOMETRY REVIEW
Calculus Appendix B
ANGLES
Radian measure
In calculus we usually use radian measure for angles.
Consider a circle of radius r = 1 centered at the origin (0, 0).
An angle is determined by rotating the “spoke” that initially goes from (0, 0) to (1, 0).
The radian measure of the angle is the distance traversed by the point on the
circumference of the circle if the rotation is counterclockwise, and it is the
negative of the distance traversed if the rotation is clockwise.
If a circle of radius r is used and the distance along the circumference is s,
then the radian measure is s/r.
Degree measure
One degree = 1° = 1/360 revolution, so 360° = 2 π radians.
Therefore 1° = π/180 ≈ .017 radian, and 1 radian = 180/π ≈ 57 degrees.
Table B.1, p. A14, lists some special angles.
Practice problems
1. Express these angles in radians and degrees:
a. 1/4 clockwise revolution
b. 1/8 clockwise revolution
c. 1/3 counterclockwise revolution
d. 2/3 counterclockwise revolution
2. Convert these degree measures into radians
a. 75°
b. 45°
c. 390°
3. Convert these radian measures into degrees
a. π/6
b. π/4
c. π/3
d. 2π/3
RIGHT TRIANGLE TRIGONOMETRY
Refer to Figure B.6, p. A15,
showing a right triangle with angle θ, adjacent leg x, opposite leg y, and hypotenuse r.
Each of the six nontrivial ratios of lengths defines a trigonometric function.
Definitions of the six trigonometric functions
4. sin(θ) =
5. cos(θ) =
6. tan(θ) =
7. cot(θ) =
8. sec(θ) =
9. csc(θ) =
SOHCAHTOA
Special triangles: Figure B.7
10.
θ
π/6 π/4 π/3
sin(θ)
cos(θ)
tan(θ)
TRIGONOMETRY of CIRCULAR FUNCTIONS
If the angle of radian measure θ corresponds to the point (x, y) on a circle of radius r
centered at the origin,
then the above definitions of the six trigonometric functions still hold.
(B.1 DEFINITION, p. A16)
Practice exercises
11. Find the values of all six trigonometric functions if (x, y) = (3, 4).
12. Find the values of all six trigonometric functions if cos(θ) = 3/5 and 0 < θ < π/2.
TRIGONOMETRIC IDENTITIES
The appendix lists more than forty trigonometric identities.
In Calculus I we’ll use mainly the Pythagorean identities (11) etc.,
the odd/even identities (16, 19, 22), the periodic identities (23, 24, 27),
and the addition formulas (34, 35) for sine and cosine.
Calculus II uses the other identities.