Steve S. answered • 02/26/14

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Tutoring in Precalculus, Trig, and Differential Calculus

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

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)