4.3 Exercises

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Exercise 4.3:1

Determine the angles \displaystyle \,v\, between \displaystyle \,\displaystyle \frac{\pi}{2}\, and \displaystyle \,2\pi\, which satisfy

a) \displaystyle \cos{v}=\cos{\displaystyle \frac{\pi}{5}} b) \displaystyle \sin{v}=\sin{\displaystyle \frac{\pi}{7}} c) \displaystyle \tan{v}=\tan{\displaystyle \frac{2\pi}{7}}
Answer
Answer 4.3:1
Solution a
Solution 4.3:1a
Solution b
Solution 4.3:1b
Solution c
Solution 4.3:1c

Exercise 4.3:2

Determine the angles \displaystyle \,v\, between 0 and \displaystyle \,\pi\, which satisfy

a) \displaystyle \cos{v} = \cos{\displaystyle \frac{3\pi}{2}} b) \displaystyle \cos{v} = \cos{ \displaystyle \frac{7\pi}{5}}
Answer
Answer 4.3:2
Solution a
Solution 4.3:2a
Solution b
Solution 4.3:2b

Exercise 4.3:3

Suppose that \displaystyle \,-\displaystyle \frac{\pi}{2} \leq v \leq \displaystyle \frac{\pi}{2}\, and that \displaystyle \,\sin{v} = a\,. With the help of \displaystyle \,a express

a) \displaystyle \sin{(-v)} b) \displaystyle \sin{(\pi-v)}
c) \displaystyle \cos{v} 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)}
Answer
Answer 4.3:3
Solution a
Solution 4.3:3a
Solution b
Solution 4.3:3b
Solution c
Solution 4.3:3c
Solution d
Solution 4.3:3d
Solution e
Solution 4.3:3e
Solution f
Solution 4.3:3f

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)}
Answer
Answer 4.3:4
Solution a
Solution 4.3:4a
Solution b
Solution 4.3:4b
Solution c
Solution 4.3:4c
Solution d
Solution 4.3:4d
Solution e
Solution 4.3:4e
Solution f
Solution 4.3:4f

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
Answer 4.3:5
Solution
Solution 4.3:5

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
Answer 4.3:6
Solution a
Solution 4.3:6a
Solution b
Solution 4.3:6b
Solution c
Solution 4.3:6c

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
Answer 4.3:7
Solution a
Solution 4.3:7a
Solution b
Solution 4.3:7b

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 4.3:8a
Solution b
Solution 4.3:8b
Solution c
Solution 4.3:8c
Solution d
Solution 4.3:8d

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
Solution 4.3:9
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