Find the values of the trigonometric functions of t from the given information. csc(t) = 7, cos(t) < 0

Question

sin(t) = 

 

cos(t) = 

 

tan(t) = 

 

sec(t) = 

 

cot(t) = 

 

Answer 1

Find the values of the trigonometric functions of t from the given information. csc(t) = 7, cos(t) < 0

ASTC means All Students Take Calculus

which means

in Quadrant I, All trig functions are positive. None are negative.

in Quadrant II, Sine and its reciprocal (cosecant) are positive. The rest are negative.

in Quadrant III, Tangent and its reciprocal (cotangent) are positive. The rest are negative.

in Quadrant IV, Cosine and its reciprocal (secant) are positive. The rest are negative.

Since csc(t)=1/sin(t), and sin(t)=y/R,

csc(t)=R/y=7

If cos(t)<0, then t must lie in Quadrants II or III

sin(t) = 1/csc(t) = 1/7

Since sin(t)>0, t must lie only in Quadrant II where cos(t)<0 and sin(t)>0

Now we can set y=1 and R=7 so that sin(t)=1/7

Since x²+y²=R²

x=√(7²-1²)=√(49-1)=√48=2√12=4√3

cos(t) = x/R = 4√3/7 ≅ 0.9897

 

tan(t) = y/x = 1/(4√3) ≅ 0.1443

 

sec(t) = 1/cos(t) = R/x = 7/(4√3) ≅ 1.010

 

cot(t) = 1/tan(t) = 4√3 ≅ 6.928