Question

How do you use an addition or subtraction formula to write the expression as a trigonometric function of one number and then find the exact value given `cos (7pi/6) cos (-pi/ 2)-sin(7pi/6)sin(-pi/2)`?

Answer 1

Recall the formula for `cos(alpha+beta)`:
`cos(alpha+beta)=cos(alpha)*cos(beta)-sin(alpha)*sin(beta)`

Using this formula for `alpha=7pi/6` and `beta=-pi/2`, we get:

`cos(7pi/6)*cos(-pi/2)-sin(7pi/6)*sin(-pi/2)=`
`=cos(7pi/6+(-pi/2))=`
`=cos((7pi)/6-(3pi)/6)=`
`=cos((2pi)/3)=-cos(pi-(2pi)/3)=-cos(pi/3)=-1/2`

As a check, notice that
`cos(7pi/6)=-cos(pi/6)=-sqrt(3)/2`,
`cos(-pi/2)=0`,
`sin(7pi/6)=-sin(pi/6)=-1/2`,
`sin(-pi/2)=-1`.
Performing direct calculations, we get the same `-1/2`.
Check!