... Given the functions (sin?, cos?, sin? and cos ?), we seek formulas that express sin(?+?) and cos(?+ ... As shown in the drawing, to derive the formula we combine two right-angled triangles . ... The triangle ADF is right-angled and has the angle (?+ ... Each of them is one of the two sharp ("acute") angles in its own right-angled triangle. Since the sharp angles in such a triangle add up to 90 degrees, the other two sharp angles must be equal. ... In the right-angled triangle CED DE = DC cos ? ...

Given the functions (sin a , cos a , sin b and cos b ), we seek formulas that express sin( a+b ) and cos( a+b ). ... As shown in the drawing, to derive the formula we combine two right-angled triangles . ABC which has an angle a . ... AB = AC cos a = R cos b cos a The triangle ADF is right-angled and has the angle ( a+b ). ... Each of them is one of the two sharp ("acute") angles in its own right-angled triangle. ... In the right-angled triangle CED DE = DC cos a = R sin b cos a . ...

Luna and Io - A Comparison Study . ... They will construct paper scale models of Io , Luna , Earth and Jupiter and use these models to explain the differences between the two moons. ... Explain to the students that the picture is not to scale, but that it represents Jupiter and one of its moons. ... Have the students divide into groups of three or four and construct scale models of the Earth/Luna system, and the Jupiter/Io system, using the facts on the data sheets. ... Jupiter, Io, Earth, and Luna . ...

Cosmic Research, Vol. 42, No. 5, 2004, pp. 435­488. Translated from Kosmicheskie Issledovaniya, Vol. 42, No. 5, 2004, pp. 453­508. Original Russian Text Copyright © 2004 by Veselovsky, Panasyuk, Avdyushin, Bazilevskaya, Belov, Bogachev, Bogod, Bogomolov, Bothmer, Boyarchuk, Vashenyuk, Vlasov, Gnezdilov, Gorgutsa, Grechnev, Denisov, Dmitriev, Dryer, Yermolaev, Eroshenko, Zherebtsov, Zhitnik, Zhukov, Zastenker, Zelenyi, Zeldovich, Ivanov-Kholodnyi, Ignat'ev, Ishkov, Kolomiytsev, Krasheninnikov, Kudela,