If sin(𝛼) = 20/29 where 0 < 𝛼 < 𝜋 /2 and cos(𝛽) = 8/17 where 3𝜋/2 < 𝛽 < 2𝜋, find the exact values of the following. (a) sin(𝛼 + 𝛽) (b) cos(𝛼 − 𝛽) (c) tan(𝛼 − 𝛽)

Question

please explain the steps you went through if you can

Answer 1

First of all, recognize that you are being asked about 2 right triangles: the first has sides 20, 21 and 29 and the second has sides 8, 15 and 17. You should verify that fact from the Pythagorean theorem.

Once you have done that you can easily compute the trig functions of α and β.

Then you need the angle addition formulae:

sin(a±b)=sin a cos b ± cos a sin b

cos(a+b)=cos a cos b - sin a sin b

cos(a-b)=cos a cos b + sin a sin b

tan(a-b)=(tan a - tan b)/(1+tan tan b)

If those formulae are new to you you should look them up either in your textbook or an old-fashioned textbook of trig.