Trig Riddle #2: I am the point
Guidance
Any point in the coordinate plane can be represented by its angle of rotation and radius, or distance from the origin. The point is said to lie on the terminal side of the angle. We can find the measure of the reference angle using right triangle trigonometry. When the point is identified in this manner we call the coordinates Polar coordinates. They are written as
Example A
Find the angle of rotation (in degrees) and radius (distance from the origin) of the point
Solution:
First, make a sketch, plot the point and drop a perpendicular to the
From the sketch, we can see that
The radius or distance from the origin is the hypotenuse of the right triangle.
r^2 &=45 \\
r &=\sqrt{45}=3\sqrt{5}
Using this information, we can write the point
Example B
Write the Cartesian coordinates,
Solution: Again, start with a sketch.
We can find the reference angle again using tangent:
Now find the radius:
r^2 &=25 \\
r &=\sqrt{25}=5
The Polar coordinates are thus
Note:
You may have noticed that there is a pattern that gives us a short cut for finding the Polar coordinates for any Cartesian coordinates,
The reference angle can be found using,
Example C
Given the point
Solution: Make sure your calculator is in radian mode. Using the shortcut, we can find the Polar coordinates:
We already know that
Now we can use the hypotenuse,
Concept Problem Revisit
First, make a sketch, plot the point and drop a perpendicular to the
From the sketch, we can see that
The radius or distance from the origin is the hypotenuse of the right triangle.
r^2 &=10 \\
r &=\sqrt{10}
Therefore, my polar coordinates are
Guided Practice
1. Find the angle of rotation (in degrees) and radius (distance from the origin) of the point
2. Write the Cartesian coordinates,
3. Given the point
Answers
1.
2.
The six trigonometric ratios are:
\cos 1.08 &=\frac{8}{17} \quad \sec 1.08 =\frac{8}{15} \\
\tan 1.08 &=\frac{15}{8} \quad \ \ \cot 1.08 =\frac{8}{15}
3.
The six trigonometric ratios are:
\cos 341.6^\circ &=\frac{3 \sqrt{10}}{10} \quad \csc 341.6^\circ=\frac{\sqrt{10}}{3} \\
\tan 341.6^\circ &=\frac{1}{3} \quad \ \ \ \tan 341.6^\circ=3
Explore More
Angle measures should be rounded to the dearest degree or hundredth of a radian or given exactly if possible. All values of
Write the following Cartesian coordinate pairs in Polar form. Use degrees for problems 1 and 2 and radians for problems 35.

(16,−30) 
(5,5) 
(−5,−12) 
(−9,40) 
(−4,8)
Given the points on the terminal side of an angle, find the Polar coordinates (in degrees) of the point and the six trigonometric ratios for the angles.

(−6,8) 
(0,−15) 
(10,−8) 
(43√,4) 
(−6,6)
Given the points on the terminal side of an angle, find the Polar coordinates (in radians) of the point and the six trigonometric ratios for the angles.

(−9,0) 
(13,−13) 
(2,3) 
(−7,−73√) 
(−8,−4)