9.5  Finding Trig Functions

  We already looked at how to find trig functions when a point, P, and the quadrant are given.

 Review this process by clicking on this link.

 This lesson focuses on finding a trig function given a trig function, its value, and the quadrant.

   Example 1

      If cos θ = 2/3,  and θ is in QIV, find tan θ.

    Solution

      Before we find tan θ, we need to find the value of sin θ. 

       We know that tan θ is defined as sin θ / cos θ, and we're missing sin θ.

      sin2 θ + cos2 θ = 1            We will use the first Pythagorean Identity.

      sin2 θ + (2/3)2 = 1            We are given that cos θ = 2/3, so we will plug-in 2/3  for cos θ.

      sin2 θ + 4/9 = 9/9           After squaring 2/3, we have a denominator of 9, so we must change 1                                                 to 9/9 because we will be subtracting 4/9 from both sides.

      sin2 θ = 5/9                          

      sin θ = ±√(5/9)                Because sin θ is squared, we need to find ± square root of 5/9.

      sin θ = - √(5/9)               Now is the time to look at which quadrant cos is located -->  QIV. 

                                            According to our previous lessons, we know that in QIV, sin θ is                                                         negative.

  We can now find tan θ using the Quotient Identity.

Example 2 

       If sin θ = -15/17, and theta is in QIII, find sec θ.

 Solution

      We know that sec θ is the reciprocal function of cos θ.  So, we need to find cos θ.  

       sin2 θ + cos2 θ = 1

     (-15/17)2 + cos2 θ = 1

      225/289 + cos2 θ = 289/289

              cos2 θ = 64/289

              cos θ = ±√(64/289)    is in QIII, so cos θ is negative

              cos θ = -8/17

     We need to find sec θ, which we determined was the reciprocal of cos θ.

      sec θ = -17/8