Exam (elaborations)
Trigonometric Functions and Their Derivatives solved questions
- Course
- Institution
Trigonometric Functions and Their Derivatives solved questions
[Show more]
Seller
Follow
jureloqoo
Content preview
CHAPTER 10Trigonometric Functions and
Their Derivatives
10.1 Define radian measure, that is, describe an angle of 1 radian.
Consider a circle with a radius of one unit (Fig. 10-1). Let the center be C, and let CA and CB be two radii
for which the intercepted arc AB of the circle has length 1. Then the central angle /LACE has a measure of one
radian.
Fig. 10-1
10.2 Give the equations relating degree measure and radian measure of angles.
I 2-rr radians is the same as 360 degrees. Hence, 1 radian = 180/Tr degrees, and 1 degree = 77/180
radians. So, if an angle has a measure of D degrees and/? radians, then D = (180/7r).R and R = (77/180)D.
10.3 Give the radian measure of angles of 30°, 45°, 60°, 90°, 120°, 135°, 180°, 270°, and 360°.
I We use the formula R = (?r/180)D. Hence 30° = 77/6 radians, 45° = 77/4 radians, 60° = 77/3 radians,
90° = 77/2 radians, 120° = 27T/3 radians, 135° = 377/4 radians, 180° = 77 radians, 270° = 377/2 radians,
360° = 277 radians.
10.4 Give the degree measure of angles of 377/5 radians and 577/6 radians.
I We use the formula D = (180 ITT)R. Thus, 377/5 radians = 108° and 577/6 radians = 150°.
10.5 In a circle of radius 10 inches, what arc length along the circumference is intercepted by a central angle of 77/5
radians?
I The arc length s, the radius r, and the central angle 6 (measured in radians) are related by the equation
s = r6. In this case, r = 10 inches and 0 = 77/5. Hence, 5 = 277 inches.
10.6 If a bug moves a distance of 377 centimeters along a circular arc and if this arc subtends a central angle of 45°, what
is the radius of the circle?
I s = rO. In this case, s = 3ir centimeters and 0 = 77/4 (the radian measure equivalent of 45°). Thus,
377 = r • 77/4. Hence, r = 12 centimeters.
10.7 Draw a picture of the rotation determining an angle of -77/3 radians.
I See Fig. 10-2. 77/3 radians = 60°, and the minus sign indicates that a 60° rotation is to be taken in the
clockwise direction. (Positive angles correspond to counterclockwise rotations.)
62
, TRIGONOMETRIC FUNCTIONS AND THEIR DERIVATIVES 63
Fig. 10-2 Fig. 10-3
10.8 Give the definition of sin 0 and cos ft
Refer to Fig. 10-3. Place an arrow OA of unit length so that its initial point O is the origin of a coordinate
system and its endpoint A is (1,0). Rotate OA about the point O through an angle with radian measure 0. Let
OB be the final position of the arrow after the rotation. Then cos 6 is defined to be the ^-coordinate of B, and
sin 0 is defined to be the ^-coordinate of B.
10.9 State the values of cos 0 and sin 0 for 0 = 0, 77/6, ir/4, ir/3, ir!2, -IT, 3ir/2, 2ir, 9ir/4.
e sin 6 cos 0
0 0 1
7T-/6 1/2 V5/2
7T/4 V2/2 V2/2
IT/3 V3/2 1/2
it 12 1 0
IT 0 -1
37T/2 -1 0
2lT 0 1
Notice that 9ir/4 = 27r+ ir/4, and the sine and cosine functions have a period of 2ir, that is, sin(fl + 2ir) =
sin 6 and cos (6 1- 277-) = cos «. Hence, sin(97r/4) = sin(7r/4) = V2/2 and cos (97T/4) = cos (Tr/4) =
V2/2.
10.10 Evaluate: (a)cos(-ir/6) (b) sin (-7T/6) (c) cos(27r/3) (d) sin (2ir/3)
(a) In general, cos (-0) = cos ft Hence, cos (-ir/6) = cos (77/6) = V5/2. (*) In general,
sin(-0)= -sin ft Hence, sin(-ir/6) = -sin (ir/6) = -|. (c) 2ir/3 = ir/2 + ir/6. We use the identity
cos (0 + ir/2) = -sin ft Thus, cos(2ir/3)= -sin (-rr/6) = -\. (d) We use the identity sin (0 + ir/2) =
cos ft Thus, sin(27r/3) = cos(7r/6) = V3/2.
10.11 Sketch the graph of the cosine and sine functions.
We use the values calculated in Problem 10.9 to draw Fig. 10-4.
10.12 Sketch the graph of y = cos 3*.
Because cos 3(* + 2tr/3) = cos (3>x + 2ir) = cos 3x, the function is of period p = 2ir/3. Hence, the
length of each wave is 277/3. The number/of waves over an interval of length 2ir is 3. (In general, this number
/, called the frequency of the function, is given by the equation /= 2ir/p.) Thus, the graph is as indicated in
Fig. 10-5.
10.13 Sketch the graph of y = 1.5 sin 4*.
The period p = ir/2. (In general, p = 2ir/b, where b is the coefficient of x.) The coefficient 1.5 is the
amplitude, the greatest height above the x-axis reached by points of the graph. Thus, the graph looks like Fig.
10-6.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jureloqoo. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.04. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67474 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now