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4.3 Exercises

From Förberedande kurs i matematik 1

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Current revision

       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

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