PRECALCULUS HELP

Question

Suppose that sec α= −5/3

 and π/2 < α < π

.

Find the exact values of cos α/2

 and tan α/2

. 

cos α/2	=
tan α/2	=

Answer 1

Hope this helps!

Answer 2

If sec α = -5/3, then cos α = 1/sec(α) = -3/5

Recall the half-angle formula cos(α/2) = ±√[(1+cos α)/2], thus:

cos(α/2) = ±√[(1-3/5)/2] = ±√[(2/5)/2] = ±√(1/5)

Between π/2 < α < π is Quadrant II, so we need to consider only -√(1/5) because cos(α) is negative there.

Also recall the half angle formula tan(α/2) = ±√[(1-cos α)/(1+cos α), thus:

tan(α/2) = ±√[(1-(-3/5))/(1-3/5)] = ±√[(8/5)/(2/5)] = ±√(8/2) = ±√4 = ±2

Again, -2 is correct here because tan(α) is negative which comes from cos(α) being negative in Quadrant II.

Thus, the exact values are cos(α/2) = -√(1/5) and tan(α/2) = -2

Hope this helped!

Answer 3

a = arcsec(-5/3)

a = (arcsec(-5/3))/2

cos(a/2) = cos((arcsec(-5/3))/2)

tan(a/2) = tan((arcsec(-5/3))/2)

Those are the exact values.

There are no exact values that can be expressed as a fraction like sin(pi/3) = 1/2.

arcsec(-5/3) = about 2.2 radians or 126.9 degrees