Question

Use the given information to find the exact value of each of the following. a. sine 2 theta b. cosine 2 theta c. tangent 2 theta sine theta equals five sixths comma theta lies in quadrant II

Answer 1

(a)
`sin 2\theta = -(5√(11) )/(18)`

(b)
`cos 2\theta= -(7)/(18)`

(c)
`tan 2\theta=`
`(5√(7) )/(11)`

Explanation:

If
`sin \theta =(5)/(6) , 90\leq \theta\leq 180`

Using Pythagoras,

Opposite=5, Hypotenuse =6, Adjacent=?

`6^2=5^2+Adj^2\\Adj^2=36-25=11\\Adjacent=√(11)`

In the Second Quadrant,

`sin \theta =(5)/(6) , cos \theta =-(√(11) )/(6), Tan \theta =-(5 )/(√(11))`

(a)
`sin 2\theta=2sin\theta cos\theta=2 X (5)/(6) X -(√(11) )/(6) = -(5√(11) )/(18)`

(b)
`cos 2\theta= cos^2\theta-sin^2\theta=(-(√(11) )/(6))^2-((5)/(6))^2= -(7)/(18)`

(c)
`tan 2\theta=(2tan\theta)/(1-tan^2\theta)`

`tan 2\theta=(2(-(5 )/(√(11))))/(1-(-(5 )/(√(11)))^2) =(-(10 )/(√(11)))/(1-(25)/(11)) =(-(10 )/(√(11)))/(-(14)/(11) )=(5√(7) )/(11)`