Trigonometry

Thinking in Polar

The graph x2+y2=9x^2 + y^2 = 9 can be considered the set of all solutions to the Pythagorean theorem a2+b2=c2,a^2 + b^2 = c^2, where cc is set at 3. Several points on the graph (with approximations) are shown above. Also note:

(1.10)2+(2.79)29(2.46)2+(1.72)29(2.12)2+(2.12)29\begin{aligned} (-1.10)^2 + (2.79)^2 &\approx 9 \\ (2.46)^2 + (1.72)^2 &\approx 9 \\ (-2.12)^2 + (-2.12)^2 &\approx 9 \end{aligned}

What does the entire graph of x2+y2=9 x^2 + y^2 = 9 look like?

               

Thinking in Polar

In a traditional xy xy-coordinate system from the previous problem, the first value xx indicates horizontal distance and the second value yy indicates vertical distance. In a polar coordinate system using (r,θ), (r, \theta), the value rr gives "distance forward from origin when facing angle θ. \theta." Some example points are shown above.

One curious thing is that rr can be negative. Which point is (3,135)? (-3, 135^\circ) ?

               

Thinking in Polar

We saw the graph of the circle above written as x2+y2=9 x^2 + y^2 = 9 in rectangular coordinates. How can you write it using polar coordinates?

               

Thinking in Polar

Other than the same point being identified with a positive rr (facing forward) or a negative rr (facing backward), it's possible for the same point to be identified in an infinite number of ways via the measure of angle.

The point below can be identified as (2,180). (2, 180^\circ) . Which of the choices also identifies the same point?

               

Thinking in Polar

In the visualization above, rr is restricted to only negative values. It can still be used to help to answer the question "Which pair of coordinates refer to the same point?"

               

Thinking in Polar

Because of possible negative values, polar graphs are often restricted to particular values of rr and θ. \theta .

If we restrict θ \theta to positive values only, what kind of graph will r=θ r= \theta be?

               

Thinking in Polar

The visualization below lets you see the effect of bb in the spiral graph r=bθ. r = b \theta .

What will happen to the graph if bb is allowed to be negative?

               

Thinking in Polar

Polar coordinates give the flexibility to include whole new classes of graphs, like the flower with overlapping petals above. Many other marvels await: start now with an intuitive look at the six main functions of trigonometry.

               
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