Processing Math: 81%
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.
No jsMath TeX fonts found -- using image fonts instead.
These may be slow and might not print well.
Use the jsMath control panel to get additional information.
jsMath Control PanelHide this Message

jsMath

4.3 Exercises

From Förberedande kurs i matematik 1

Jump to: navigation, search
       Theory          Exercises      

Exercise 4.3:1

Determine the angles v between 2 and 2 which satisfy

a) cosv=cos5 b) sinv=sin7 c) tanv=tan72

Answer | Solution a | Solution b | Solution c

Exercise 4.3:2

Determine the angles v between 0 and which satisfy

a) cosv=cos23 b) cosv=cos57

Answer | Solution a | Solution b

Exercise 4.3:3

Suppose that −2v2 and that sinv=a. With the help of a express

a) sin(−v) b) sin(−v)
c) cosv d) sin2−v 
e) cos2+v  f) sin3+v 

Answer | Solution a | Solution b | Solution c | Solution d | Solution e | Solution f

Exercise 4.3:4

Suppose that 0v and that cosv=b. With the help of b express

a) sin2v b) sinv
c) sin2v d) cos2v
e) sinv+4  f) cosv−3 

Answer | Solution a | Solution b | Solution c | Solution d | Solution e | Solution f

Exercise 4.3:5

Determine cosv and tanv, where v is an acute angle in a triangle such that sinv=75.

Answer | Solution

Exercise 4.3:6

a) Determine  sinv  and  tanv  if  cosv=43  and  23v2.
b) Determine  cosv  and  tanv  if  sinv=310  and v lies in the second quadrant.
c) Determine  sinv  and  cosv  if  tanv=3  and  v23.

Answer | Solution a | Solution b | Solution c

Exercise 4.3:7

Determine  sin(x+y)  if

a) sinx=32, siny=31  and x, \displaystyle \,y\, are angles in the first quadrant.
b) \displaystyle \cos{x}=\displaystyle \frac{2}{5}\,, \displaystyle \ \cos{y}=\displaystyle \frac{3}{5}\ and \displaystyle \,x\,, \displaystyle \,y\, are angles in the first quadrant.

Answer | Solution a | Solution b

Exercise 4.3:8

Show the following trigonometric relations

a) \displaystyle \tan^2v=\displaystyle\frac{\sin^2v}{1-\sin^2v}
b) \displaystyle \displaystyle \frac{1}{\cos v}-\tan v=\frac{\cos v}{1+\sin v}
c) \displaystyle \tan\displaystyle\frac{u}{2}=\frac{\sin u}{1+\cos u}
d) \displaystyle \displaystyle\frac{\cos (u+v)}{\cos u \cos v}= 1- \tan u \tan v

Solution a | Solution b | Solution c | Solution d

Exercise 4.3:9

Show Feynman's equality
\displaystyle \cos 20^\circ \cdot \cos 40^\circ \cdot \cos 80^\circ = \displaystyle\frac{1}{8}\,\mbox{.}
(Hint: use the formula for double angles on \displaystyle \,\sin 160^\circ\,.)

Solution

Retrieved from "http://wiki.math.se/wikis/2009/bridgecourse1-ImperialCollege/index.php/4.3_Exercises"