Trigonometric Ratios I and II

Trigonometric Ratios Part One 

If P(θ)=(x,y) is the point on the terminal arm of angle θ that intersects the uni circle, notice that... 

Where:                                                                                         Reciprocal functions:

cosθ= adjacent/hypotenuse   →  cosθ= Y        →   secθ= 1/X or hypotenuse/adjacent

sinθ= opposite/hypotenuse   →  sinθ= X        →   cscθ=1/Y or hypotenuse/adjacent

tanθ= opposite/adjacent        →  tanθ= Y/X   →   cotθ=X/Y or adjacent/hypotenuse

or SOH CAH TOA 

Recall that the unit circle is a circle with a radius of 1, centre at (0,0) and equation 
x^2+y^2=1. You can now describe the equation for the unit circle as cos^2θ+sin^2θ=1

Exact Values and Approximate Values of Trigonometric Ratios 

• Exact Values for the trigonometric ratios can be determined using special triangles 30°,60°,45°... and multiples of θ= 0, π/6,π/4,π/3... for points P(θ) on the unit circle. 
• Approximate Values can determine using scientific calculator. Set the mode first to Degree (D) or Radian (R) to correct the measure. 

Examples:

The point A(3/5,4/5) lies at the intersection of the unit circle and the terminal arm of an angle θ in standard position. 

a) Draw a diagram to model the situation.
b) Determine the values of the six trigonometric ratios for θ, in lowest form. 

*Recall the CAST Rule and SOH CAH TOA

a)

The diagram is in Quadrant I, where all the trigonometric ratios are positive. 

b) cosθ= adj./hyp.= 3/5                           secθ= hyp./adj.= 5/3
     sinθ= opp./hyp.= 4/5                          cscθ= hyp./opp.= 5/4
     tanθ= opp./adj.= 4/3                           cotθ= adj./opp.= 3/4 

Determine the exact value. 
a) cos 5π/6 = -√3/2 
because 5π/6 is a special triangle and it is negative because it lies in Quadrant II . 

Determine the approximate value.  
a) cos 260°=
Solution: 
 In degree mode, 
cos(260 = -0.1736481777 (answer) 

Trigonometric Ratios Part Two 

 Approximate Values of Angle
- To determine θ, use the inverse trigonometric function key on calculator. 

Example:

a) sinθ= 0.879 in the domain 0≤θ≤2π. 
Solution:
    *Recall, CAST Rule. Then, determine which Quadrant is going to lie. Since sinθ is positive it will lies in Quadrant I and II. Using calculator enter the inverse sine

sinθ= 0.879
         sin^{−1 (0.879}
            θR= 1.073760909
QI θ=θR                                                           QII θ=π-θR
       θ=1.073760909                                                  π-1.073760909
                                                                                    θ=2.067831744

Answer: θ= 1.073760909, 2.067831744 

THE END :)) 

and I choose.....

 Cathlene☺