Evaluate csc − 1 ⁡ ( sec ⁡ 5 ) without using a calculator.

Question

Evaluate       csc          −      1        ⁡  (  sec  ⁡  5  ) without using a calculator.

Nola Rivera · Accepted Answer

csc          −      1        ⁡  sec  ⁡  5  =  θ  sec  ⁡  5  =  csc  ⁡  θ      hypotenuse    side adjacent to 5    =  sec  ⁡  5  =  csc  ⁡  θ  =      hypotenuse          side opposite       θ        Hence:                  side adjacent to 5  =  side opposite   θSo   5 and   θ are the two non-right angles in a right triangle. They add up to a right angle:  θ  +  5  =  right angle(If this is supposed to be in degrees, then   θ  =  85.  If it&#039;s in radians, we would need the angle between   −  π  /  2 and   +  π  /  2 that has the right cosecant.)

Reed Eaton · Accepted Answer

Let   α  =      csc          −      1        ⁡      (    sec    ⁡    5    )    . In other words,  is an angle in the first or fourth quadrant, and   csc  ⁡  α  =  sec  ⁡  5. Put another way,   sin  ⁡  α  =  cos  ⁡  5. Do you see why this is the same thing? Can you take it from here, perhaps by drawing a right triangle? Note: You will be able to find an exact value this way. It will not be a numerical answer.

Evaluate csc − 1 </mrow> </msup> ⁡<

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