Question

cosx=-root2/3 and x is in quadrant 3

Use identities to find the exact values at x for the remaining five trigonometric functions

Answer 1

We can start by drawing a reference triangle in quadrant 3, where cos(x) is negative and equal to -√2/3. This will be a 30-60-90 triangle with hypotenuse 1 and opposite side √3/2.

The adjacent side is negative since cos(x) is negative, so we have:

cos(x) = -√2/3 = adjacent/hypotenuse = -1/√3

multiplying both sides by -√3, we get:

-√2 = adjacent

And:

sin(x) = opposite/hypotenuse = √3/2

tan(x) = opposite/adjacent = -√3/2

csc(x) = 1/sin(x) = 2/√3

sec(x) = 1/cos(x) = -√3/2

cot(x) = 1/tan(x) = -2/√3