find the exact values of the following:

Question

1. sec30° 
2. cos75° 
3. cot 2π/3 
4. cos π/8

Answer 1

1). sec 30⁰ .

• π/6 is in quadrant 1 and there is no need for either coterminal or reference angles.

• In quadrant 1 the secant is positive.

• So, sec (π/6) is evalueated directly as follows.

           sec (π/6) = 2/√3.

2). cos 75⁰ .

• 75⁰ is in quadrant 1.in quadrant 1 the cosin is positive.

          hence

• cos (75⁰) = cos (45⁰ + 30⁰)

                         = cos 45 cos 30 - sin 45 sin 30                                   ( cos (A + B) = cos A cos B - sin A sin B)

                         = (1/√2)(√3/2) - (1/√2)(1/2)

                         = (√3 - 1)/2√2.

3). cot (2π/3).

• 2π/3 is in second quadrant.in second quadrant the tangent and cotangent are negative.

           hence

• cot (2π/3) = cot (π - 2π/3) = - cot (π/3) = - 1/√3.

4). cos (π/8).

• π/8 is in quadrant 1 and there is no need for either coterminal or reference angles.
• In quadrant 1 the cosin is positive.

            hence

• cos(2(π/8)) = 2 cos^2(π/8) - 1                                           (cos (2x) = 2cos^2 (x) - 1)

           ⇒ cos (π/8) = √{ [ cos(2(π/8)) + 1 ]/2 }

                             = √{ [ cos(π/4) + 1 ]/2 }

                             = √{ [√2/2 + 1 ]/2 }

                             = √( 2+√2)/2

                             = 0.9238.