Find the exact values of the trig functions.

Question

Let P be the point on the unit circle U that corresponds to t. Find the coordinates of P and the exact values of the trigonometric functions of t, whenever possible. (If an answer is undefined, enter UNDEFINED.)

 

n=pie

 

(a) t = 5π/2
P(x, y)=



sin(5π/2)=
cos(5π/2)=
tan(5π/2)=
csc(5π/2)=
sec(5π/2)=
cot(5π/2)=

 

(b) t = −π/2
P(x, y)=



sin(−π/2)=
cos(−π/2)=
tan(−π/2)=
csc(−π/2)=
sec(−π/2)=
cot(−π/2)=

Answer 1

Hey Colsten,

 

5π/2 = 2π + π/2 = π/2

 

 

 

sin   cos   tan   csc   sec   cot

 1      0    und    1    und    0

               ∞             ∞

-π/2  = 3π/2        -π/2 = 0-π/2    3π/2 = (π + π/2)  or (2π - π/2)

 

sin   cos   tan   csc   sec   cot

 -1    0     und   -1   und     0

               -∞           ∞

 

 

 

 

Answer 2

I am assuming angle measurements are in radians.  :)

First, the trig functions are periodic, with periodicity of 2π.  So, we can subtract this from (5π/2): (5π/2) - 2π = (π/2).

Thus, the trig function of (5π/2) is the same as the trig function of (π/2), which we are supposed to memorize (hint - if you haven't, please do so).

 

sin(5π/2)= 1
cos(5π/2)= 0
tan(5π/2)= Undefined
csc(5π/2)= 1
sec(5π/2)= Undefined
cot(5π/2)= 0

 

Now, the coordinates of the point are the radius of the circle times the cosine of the angle for x, then the radius of the circle times the sine of the angle for y.  The radius of the unit circle is 1, so the coordinates at (5π/2) are just (0,1).

 

For the second part, (−π/2), all we are doing is going one quarter the way around the unit circle, but in a clockwise direction.  Thus, the point is now on the negative y-axis, and its coordinates are (0,-1)

 

sin(−π/2)= -1
cos(−π/2)= 0
tan(−π/2)= Undefined
csc(−π/2)= -1
sec(−π/2)= Undefined
cot(−π/2)= 0

 

 

I would like to point out that the sine is the ratio of the y-coordinate of the point on the circle to the radius of that circle:

 

sinθ=(y/r)

 

Similarly, the cosine is x/r

cosθ=(x/r)

 

The tangent is just the sine over the cosine:

 

tanθ = sinθ / cosθ = (y/r)/(x/r) = (y/x)

 

The cotangent is the multiplicative inverse of the tangent, the secant is the multiplicative inverse of the cosine, and the cosecant is the multiplicative inverse of the sine.  So, we have this pattern:

 

sinθ = (y/r)    cscθ = (r/y)

cosθ = (x/r)    secθ = (r/x)

tanθ = (y/x)    cotθ = (x/y)

 

I write them down in a column of three, then up in the next column.  Crossing over from the first column to the second, I just flip the fractions over to remember what the ratios are.  This helps with problems like the one we have here, and it helps me when I am doing trig identities.