4.2 Trigonometric Functions: The Unit Circle Back in chapter one, we discussed the circles. The unit circle is given by 2 2 1 x y because the radius is one. We can define all six trig functions using this unit circle. The trig functions are sine, cosine, tangent, cotangent, secant, and cosecant. Definitions of Trigonometric Functions: Let t be a real number and let (x, y) be a point on the unit circle corresponding to t. (We picture the real number line wrapped around the unit circle.) sin t y cos t x tan , 0 y t x x 1 csc , 0 t y y 1 sec , 0 t x x cot , 0 x t y y In section 1 of chapter 4 we looked at converting between radians and degrees. Sometimes, having things memorized will be in your best interest. I encourage you to memorize the following table. I’ll explain why in class. Degrees Radians sin t cos t tan t 0 0 0 0 2 4 1 2 0 30 π/6 1 1 2 2 3 2 1/2 1 3 3 3/2 3 45 π/4 2 2 2 2 2/2 1 2/2 60 π/3 3 2 1 1 2 2 3/2 3 1/2 90 π/2 4 2 1 2 2 0 0 2 undefined
Transcript
4.2 Trigonometric Functions: The Unit Circle
Back in chapter one, we discussed the circles. The unit circle is given by 2 2 1x y because the
radius is one. We can define all six trig functions using this unit circle. The trig functions are sine,
cosine, tangent, cotangent, secant, and cosecant.
Definitions of Trigonometric Functions: Let t be a real number and let (x, y) be a point on the unit
circle corresponding to t. (We picture the real number line wrapped around the unit circle.)
sin t y cost x tan , 0y
t xx
1
csc , 0t yy
1
sec , 0t xx
cot , 0x
t yy
In section 1 of chapter 4 we looked at converting between radians and degrees. Sometimes, having
things memorized will be in your best interest. I encourage you to memorize the following table. I’ll
explain why in class.
Degrees Radians sin t cos t tan t
0 0 00
2
41
2
0
30 π/6 1 1
2 2
3
2
1/ 2 1 3
33 / 2 3
45 π/4 2
2
2
2
2 / 21
2 / 2
60 π/3 3
2
1 1
2 2
3 / 23
1/ 2
90 π/2 4 21
2 2
00
2
undefined
We can divide the unit circle up into several pieces and find the values of the six functions. It would
be in your best interest to memorize this information. You will need it in calculus and every other
higher math class you take from this day forward.
sin t y cost x tan , 0y
t xx
1csc , 0t y
y
1sec , 0t x
x cot , 0
xt yy
Examples: Find the point (x,y) on the unit circle that corresponds to the real number t.
1. t 2. / 3t 3. 5 /3t
Examples: Evaluate, if possible, the sine, cosine, and tangent of the real number.
1. / 4t
2. / 6t
3. 7 / 4t
4. 2 / 3t
Examples: Evaluate, if possible, the six trig functions of the real number.
1. 5 / 6t
2. 3 / 4t
3. / 2t
Domain of Periodic Function: A function f is periodic if there exists a positive real number c such that
f t c f t for all t in the domain of f. The smallest number c for which f is periodic is called the
period of f.
Fact: Both sine and cosine have a period of 2π.
Even and Odd Trig Functions:
The cosine and secant functions are even so, cos cost t and sec sect t .
The other four trig functions are odd so, sin sint t , tan tant t ,
