Problems from Trigonometric identities: half an angle, double an angle, sum and difference of two angles

Calculate the trigonometric ratios of the angle of 15 degrees in terms of the angles of 45 degrees and 30 degrees.

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Development:

First of all, we observe that 15=4530. Therefore, knowing the ratios of 45 degrees and 30 degrees, we will be able to obtain those of the 15 degree angle. Let's calculate: sin(45)=22    sin(30)=12 cos(45)=22    cos(30)=32 tan(45)=1    tan(30)=33

From the following formulas, we have:

sin(15)=sin(4530)=sin(45)cos(30)cos(45)sin(30)= =22322212=23424=24(31)

cos(15)=cos(4530)=cos(45)cos(30)+sin(45)sin(30)= =2232+2212=234+24=24(3+1)

tan(15)=tan(4530)=tan(45)tan(30)1+tan(45)tan(30)= 1331+33=3333+33=333+3=333+33333=9+36393= =12636=23

Solution:

sin(15)=24(31)

cos(15)=24(3+1)

tan(15)=23

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