Question #29146
Sum/difference trig problems please help urgent?
1. cos(345)
2. sin(405)
3. sin(10pi/24)
4. tan(-105)
answer has to be in rads please like 2pi/4
Solution.
1. It is known that cos ( α − β ) = cos ( α ) cos ( β ) + sin ( α ) sin ( β ) \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) cos ( α − β ) = cos ( α ) cos ( β ) + sin ( α ) sin ( β )
Observe that 34 5 o = 36 0 0 − 1 5 o = 2 π − π 12 345^o = 360^0 - 15^o = 2\pi - \frac{\pi}{12} 34 5 o = 36 0 0 − 1 5 o = 2 π − 12 π cos ( 2 π ) = 1 \cos(2\pi) = 1 cos ( 2 π ) = 1 sin ( 2 π ) = 0 \sin(2\pi) = 0 sin ( 2 π ) = 0
cos ( 34 5 o ) = cos ( 2 π − π 12 ) = cos ( 2 π ) cos ( π 12 ) + sin ( 2 π ) sin ( π 12 ) = cos ( π 12 ) . \cos(345^o) = \cos\left(2\pi - \frac{\pi}{12}\right) = \cos(2\pi)\cos\left(\frac{\pi}{12}\right) + \sin(2\pi)\sin\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{12}\right). cos ( 34 5 o ) = cos ( 2 π − 12 π ) = cos ( 2 π ) cos ( 12 π ) + sin ( 2 π ) sin ( 12 π ) = cos ( 12 π ) .
Taking into account that cos ( α ) = 1 + cos ( α ) 2 \cos(\alpha) = \sqrt{\frac{1 + \cos(\alpha)}{2}} cos ( α ) = 2 1 + c o s ( α ) cos ( π 6 ) = 3 2 \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} cos ( 6 π ) = 2 3
cos ( 34 5 o ) = cos ( π 12 ) = 1 + 3 2 2 = 2 + 3 2 \cos(345^o) = \cos\left(\frac{\pi}{12}\right) = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \frac{\sqrt{2 + \sqrt{3}}}{2} cos ( 34 5 o ) = cos ( 12 π ) = 2 1 + 2 3 = 2 2 + 3
Answer. 2 + 3 2 \frac{\sqrt{2 + \sqrt{3}}}{2} 2 2 + 3
2. Since sin ( α + β ) = sin ( α ) cos ( β ) + cos ( α ) sin ( β ) \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) sin ( α + β ) = sin ( α ) cos ( β ) + cos ( α ) sin ( β )
sin ( 40 5 o ) = sin ( 36 0 o + 4 5 o ) = sin ( 2 π + π 4 ) = sin ( 2 π ) cos ( π 4 ) + sin ( π 4 ) cos ( 2 π ) = sin ( π 4 ) = 2 2 \sin(405^o) = \sin(360^o + 45^o) = \sin\left(2\pi + \frac{\pi}{4}\right) = \sin(2\pi)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right)\cos(2\pi) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} sin ( 40 5 o ) = sin ( 36 0 o + 4 5 o ) = sin ( 2 π + 4 π ) = sin ( 2 π ) cos ( 4 π ) + sin ( 4 π ) cos ( 2 π ) = sin ( 4 π ) = 2 2
Answer. 2 2 \frac{\sqrt{2}}{2} 2 2
3. Observe that sin ( 10 π 24 ) = sin ( 12 π − 2 π 12 ) = sin ( π 2 − π 12 ) \sin\left(\frac{10\pi}{24}\right) = \sin\left(\frac{12\pi - 2\pi}{12}\right) = \sin\left(\frac{\pi}{2} - \frac{\pi}{12}\right) sin ( 24 10 π ) = sin ( 12 12 π − 2 π ) = sin ( 2 π − 12 π )
sin ( α − β ) = sin ( α ) cos ( β ) − cos ( α ) sin ( β ) \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) sin ( α − β ) = sin ( α ) cos ( β ) − cos ( α ) sin ( β )
sin ( 10 π 24 ) = sin ( π 2 − π 12 ) = sin ( π 2 ) cos ( π 12 ) − sin ( π 12 ) cos ( π 2 ) = cos ( π 12 ) = 2 + 3 2 . \sin\left(\frac{10\pi}{24}\right) = \sin\left(\frac{\pi}{2} - \frac{\pi}{12}\right) = \sin\left(\frac{\pi}{2}\right)\cos\left(\frac{\pi}{12}\right) - \sin\left(\frac{\pi}{12}\right)\cos\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2 + \sqrt{3}}}{2}. sin ( 24 10 π ) = sin ( 2 π − 12 π ) = sin ( 2 π ) cos ( 12 π ) − sin ( 12 π ) cos ( 2 π ) = cos ( 12 π ) = 2 2 + 3 .
Answer. 2 + 3 2 \frac{\sqrt{2 + \sqrt{3}}}{2} 2 2 + 3
4. It is known that tan ( − α ) = − tan ( α ) \tan(-\alpha) = -\tan(\alpha) tan ( − α ) = − tan ( α ) tan ( α + β ) = tan ( α ) + tan ( β ) 1 − tan ( α ) tan ( β ) \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)} tan ( α + β ) = 1 − t a n ( α ) t a n ( β ) t a n ( α ) + t a n ( β )
tan ( − 10 5 0 ) = − tan ( 10 5 0 ) = − tan ( π 3 + π 4 ) = − tan ( π 3 ) + tan ( π ) 1 − tan ( π 3 ) tan ( π ) = − 3 + 1 1 − 3 ⋅ 1 = − 3 + 1 1 − 3 \tan(-105^0) = -\tan(105^0) = -\tan\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = -\frac{\tan\left(\frac{\pi}{3}\right) + \tan(\pi)}{1 - \tan\left(\frac{\pi}{3}\right)\tan(\pi)} = -\frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} = -\frac{\sqrt{3} + 1}{1 - \sqrt{3}} tan ( − 10 5 0 ) = − tan ( 10 5 0 ) = − tan ( 3 π + 4 π ) = − 1 − tan ( 3 π ) tan ( π ) tan ( 3 π ) + tan ( π ) = − 1 − 3 ⋅ 1 3 + 1 = − 1 − 3 3 + 1
Answer. − 3 + 1 1 − 3 -\frac{\sqrt{3} + 1}{1 - \sqrt{3}} − 1 − 3 3 + 1
#339914 on Dec 2023