Trigonometric Functions Discussion Assignment

Trigonometric Functions Discussion Assignment
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Trigonometric Functions Discussion Assignment
Angle Measurement
For questions 383 to 392, convert the angle measurements as indicated. Recall
that because 2π = 360°, 1° = radians and 1 radian = .
383. Convert 45° to radians.
384. Convert 1 radian to degrees.
385. Convert 60° to radians.
386. Convert radians to degrees.
387. Convert 127° to radians.
388. Convert 25 radians to degrees.
389. Convert 135° to radians.
390. Convert 0° to radians.
391. Convert 28°43′25″ to decimal form. Round to two decimal places.
392. Convert 57.5692 to degree, minute, second form. Round seconds to the
nearest whole number.
Trigonometric Functions
Important concepts to recall:
1. Definitions of trigonometric (trig) functions: If an angle in the x-y plane is
in standard position (that is, its vertex is at the origin and its initial side
coincides with the positive x-axis) and if (a,b) is on the terminal side of
and r is the distance from (0,0) to (a,b) as shown in the figure below, then
the trig functions have the following definitions:
Note: From the preceding definitions, sine and cosecant are reciprocals of
each other, cosine and secant are reciprocals of each other, and tangent and
cotangent are reciprocals of each other. Therefore, it is necessary to
remember the definitions for the sine, cosine, and tangent functions only
because you can determine the other functions by using the reciprocal
relationships.
Trigonometric Functions Discussion Assignment
2. Unit circle: As shown in the figure below, the unit circle is centered at the
origin with radius r = 1. If (x, y) is a point on the unit circle, lying on the
terminal side of an angle in standard position, then the y and x coordinates
are the sine and cosine of angle , respectively. As the point (x, y) rotates
counterclockwise around the circle, values of the trig functions can be
determined. For instance, at = 0, sin = sin 0 = = 0. and cos = cos
0 = = 1; and at = , sin = sin = = 1 and cos =
= 0. Furthermore, if the terminal side of does not lie on an axis, a trig
function of is positive or negative depending on the quadrant in which the
terminal side of is located. You need to remember only in which
quadrants the sine, cosine, and tangent are positive because they are
negative in the other quadrants, and their reciprocals have the same signs as
the functions themselves do. The sine function is positive in quadrants I and
II; the cosine function is positive in quadrants I and IV; and the tangent
function is positive in quadrants I and III.
3. Reference angles: A reference angle for an angle A in standard position
(whose terminal side does not lie on an axis) is the positive angle between
the x-axis and the terminal side of the angle. The following figure depicts
∠A and its associated reference angle, ∠Ar
, as the terminal side of ∠A
rotates counterclockwise around the origin from quadrant I (a) to quadrant
II (b) to quadrant III (c) to quadrant IV (d).
Angles whose terminal sides coincide with a coordinate axis are their own
reference angles.
Trigonometric Functions Discussion Assignment
4. Right triangle trig ratios: The ratios relative to angle A in right triangle ABC
are as follows:
sine of ∠A = sin A =
cosine of ∠A = cos A =
tangent of ∠A = tan A =
cosecant of ∠A = csc A =
secant of ∠A = csc A =
cotangent of ∠A = csc A =
5. Special right triangles: Two special right triangles from which you can
calculate trigonometric ratios are shown below.
Note: Any triangles similar to these two right triangles have the same trig
ratio values.
For questions 393 to 407, find the exact value of the trigonometric function. Do
not use a calculator.
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