Pre Calculus help

Question

if cos x = 5/8 and sin x < 0, find the values of the other five elementary trigonometric at x. Draw a sketch with an appropriate reference triangle.
sin x =
tan x =
sec x =
csc x =
cot x =

Answer 1

Because cos x = 5/8 and sin x < 0, x is an angle in the fourth quadrant.

 

Use the Pythagorean identity to find sin x, now (make it negative).

 

From this, you easily obtain tangent, and by taking reciprocals, you will obtain the other three trig values.

Answer 2

Hello Jess.

 

If cos(x)>0 and sin(x)<0, which quadrant is triangle in?

 

Remember:

 

cos(θ) = x/r = x/√(x²+y²),

sin(θ) = y/r = y/√(x2+y2),

tan(θ) = y/x,

sec(θ) = 1/cos(x) = r/x,

csc(θ) = 1/sin(x) = r/y, and

cot(θ) = 1/tan(x) = x/y.

 

if cos(x) is positive and sin(x) is negative, then x is positive and y is negative, so it is in the fourth quadrant.

cos(x) = 5/8, so x = 5, r=8, and y=-√(8²-5²)=-√(39).

 

So,

sin(x) = -√(39)/8

tan(x) = -√(39)/5

sec(x) 8/5

csc(x) = -8/√(39) = -8√(39)/39 and

cot(x) = -5/√(39) = -5√(39)/39.