The value of cos (a+b) when cos a= 3/5 with 0<a< π/2 and Sin b = - 8/17 with 3π / 2 <b < 2 π, is 0.906.
What is triangle angle sum theorem?
According to the triangle angle sum theorem, the sum of all the angle(interior) of a triangle is equal to the 180 degrees. The formula for cos compound angle using angle sum theorem can be given as,
[tex]\cos (a+b)=\cos (a)\cos (b)-\sin (a)\sin (b)[/tex]
The exact value of cos (a+b) has to be find out. It is given that,
[tex]\cos a= \dfrac{3}{5}[/tex]
The value of a is lies between 0<a< π/2. The value of other side of the triangle using the Pythagoras theorem,
[tex]5^2={x^2+3^2}\\x^2=25-9\\x=\sqrt{16}\\x=4[/tex]
Thus, the value of sin a is 4/5 from the trigonometry ratio. the value of sine b is,
[tex]\sin b = -\dfrac{8}{17}[/tex]
The value of b is lies between 3π / 2 <b < 2 π. The value of the other side of the triangle using the Pythagoras theorem,
[tex]17^2={(-8)^2+y^2}\\y^2=289-64\\y=\sqrt{225}\\y=15[/tex]
Thus, the value of cos b is, 15/17. Put the values in above formula,
[tex]\cos (a+b)=\cos (a)\cos (b)-\sin (a)\sin (b)\\\cos (a+b)=\dfrac{3}{5}\times\dfrac{15}{17}-\dfrac{4}{5}\times\dfrac{-8}{17}\\\cos (a+b)=\dfrac{9}{17}+\dfrac{32}{85}\\\cos (a+b)=0.906[/tex]
Thus, the value of cos (a+b) when cos a= 3/5 with 0<a< π/2 and Sin b = - 8/17 with 3π / 2 <b < 2 π, is 0.906.
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