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4.3 Übungen

Aus Online Mathematik Brückenkurs 1

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Version vom 13:47, 10. Sep. 2008

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Exercise 4.3:1

Determine the angles v between 2 and 2 which satisfy

cosv=cos

sinv=sin

tanv=tan72

Answer | Solution a | Solution b | Solution c

Exercise 4.3:2

Determine the angles v between 0 and which satisfy

cosv=cos23

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)	\displaystyle \sin{\left(\displaystyle \frac{\pi}{2}-v\right)}
e)	\displaystyle \cos{\left( \displaystyle \frac{\pi}{2} + v\right)}	f)	\displaystyle \sin{\left( \displaystyle \frac{\pi}{3} + v \right)}

Exercise 4.3:4

Suppose that \displaystyle \,0 \leq v \leq \pi\, and that \displaystyle \,\cos{v}=b\,. With the help of \displaystyle \,b express

a)	\displaystyle \sin^2{v}	b)	\displaystyle \sin{v}
c)	\displaystyle \sin{2v}	d)	\displaystyle \cos{2v}
e)	\displaystyle \sin{\left( v+\displaystyle \frac{\pi}{4} \right)}	f)	\displaystyle \cos{\left( v-\displaystyle \frac{\pi}{3} \right)}

Exercise 4.3:5

Determine \displaystyle \,\cos{v}\, and \displaystyle \,\tan{v}\,, where \displaystyle \,v\, is an acute angle in a triangle such that \displaystyle \,\sin{v}=\displaystyle \frac{5}{7}\,.

Answer | Solution

Exercise 4.3:6

a)	Determine \displaystyle \ \sin{v}\ and \displaystyle \ \tan{v}\ if \displaystyle \ \cos{v}=\displaystyle \frac{3}{4}\ and \displaystyle \ \displaystyle \frac{3\pi}{2} \leq v \leq 2\pi\,.
b)	Determine \displaystyle \ \cos{v}\ and \displaystyle \ \tan{v}\ if \displaystyle \ \sin{v}=\displaystyle \frac{3}{10}\ and \displaystyle \,v\, lies in the second quadrant.
c)	Determine \displaystyle \ \sin{v}\ and \displaystyle \ \cos{v}\ if \displaystyle \ \tan{v}=3\ and \displaystyle \ \pi \leq v \leq \displaystyle \frac{3\pi}{2}\,.

Answer | Solution a | Solution b | Solution c

Exercise 4.3:7

Determine \displaystyle \ \sin{(x+y)}\ if

a)	\displaystyle \sin{x}=\displaystyle \frac{2}{3}\,,\displaystyle \ \sin{y}=\displaystyle \frac{1}{3}\ and \displaystyle \,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

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/4.3_%C3%9Cbungen“

3. Wurzeln und Logarithmen

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