unit circle

Question

Evaluate the six trigonometric function of real number

 t=(4)(3.14)/3

 

2)Evaluate the sine, cosine and tangent of the real number

t=(3)(3.14)/4

 

I got this to problems wrong can someone explain how to solve it. 

Answer 1

1) Evaluate the six trigonometric function of real number t=(4)(3.14)/3
2) Evaluate the sine, cosine and tangent of the real number t=(3)(3.14)/4

 

1)

 

3.14 ≈ pi radians = 180°

 

So t =4 pi/3 = 4(180°)/3 = 4(60°) = 240° which is in the 3rd Quadrant where tangent and cotangent are positive and all the others are negative.

 

The reference angle is 60° and we know from Special Triangle 30°-60°-90° that sin(60°) = √(3)/2 and cos(60°) = 1/2. So:

 

sin(240°) = -sin(60°) = -√(3)/2

cos(240°) = -cos(60°) = -1/2

tan(240°) = sin(240°)/cos(240°) = √(3)

cot(240°) = 1/tan(240°) = 1/√(3) = √(3)/3

sec(240°) = 1/cos(240°) = -2

csc(240°) = 1/sin(240°) = -2/√(3) = -2√(3)/3

 

2)

 

t=(3)(3.14)/4 = 3(180°)/4 = 3(45°) = 135° which is in the 2nd Quadrant where sine and cosecant are positive and all others negative.

 

The reference angle is 45° and from the 45°-45°-90° Special Triangle we know that sin(45°) = cos(45°) = 1/√(2) = √(2)/2.

 

So:

sin(135°) = sin(45°) = 1/√(2) = √(2)/2

cos(135°) = -cos(45°) = -1/√(2) = -√(2)/2

tan(135°) = sin(135°)/cos(135°) = -1

cot(135°) = 1/tan(135°) = -1

sec(135°) = 1/cos(135°) = -√(2)

csc(135°) = 1/sin(135°) = √(2)

Answer 2

Hi Daisy,

 If you have a "unit circle" the any ordered set of point on the circle x,y can be used to define the trig functions sin(θ) =y and cos(θ) =x and tan(θ) =y/x (likewise for the reciprocal functions csc sec cotan)

Rather than repeat lots of relationships check out page 3 of    http://www.math.uci.edu/sites/math.uci.edu/files/trig.pdf  there is a drawing or the unit circle with all the principal angles labeled (both in degrees and radians)

For your problem θ= 4π/3 or 240 degrees you can then read off the values i.e. sin(4π/3)=-√3/2, cos(4π/3)=-1/2 and tan(4π/3)=√3. 

2. for θ=3π/4 or 135 degrees you can now read off the values i.e. sin(3π/4)=√2/2 and so on...

You should learn everything on this "Cheat Sheet" it makes life much easier.

Regards

Jim S