cosx=-root2/3 and x is in quadrant 3
Use identities to find the exact values at x for the remaining five trigonometric functions
To find the exact values of the remaining trigonometric functions given that cos(x) = -√2/3 and x is in quadrant 3, we can use the Pythagorean identity and the definitions of the trigonometric functions.
1. To find sin(x), we can use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Plugging in the given value of cos(x), we have sin^2(x) + (-√2/3)^2 = 1.
Simplifying, sin^2(x) + 2/3 = 1.
Subtracting 2/3 from both sides, sin^2(x) = 1 - 2/3 = 3/3 - 2/3 = 1/3.
Taking the square root, we have sin(x) = ±√(1/3).
Since x is in quadrant 3 and sin is negative in that quadrant, we have sin(x) = -√(1/3).
2. To find tan(x), we can use the definition: tan(x) = sin(x)/cos(x).
Plugging in the values we found, tan(x) = (-√(1/3))/(-√2/3).
Simplifying, tan(x) = (√(1/3))/(√2/3).
Multiplying the numerator and denominator by √3, tan(x) = (√(1/3))/(√2/3) * (√3/√3).
Simplifying further, tan(x) = (√(1/3 * 3))/(√2/√3).
tan(x) = (√1)/(√2) = 1/√2 = √2/2.
3. To find csc(x), we can use the reciprocal definition: csc(x) = 1/sin(x).
Plugging in the value we found for sin(x), csc(x) = 1/(-√(1/3)).
Simplifying, csc(x) = -(1/√(1/3)) = -√3.
4. To find sec(x), we can use the reciprocal definition: sec(x) = 1/cos(x).
Plugging in the given value of cos(x), sec(x) = 1/(-√2/3).
Simplifying, sec(x) = -(1/(√2/3)) = -√3/√2 = -√6/2.
5. To find cot(x), we can use the reciprocal definition: cot(x) = 1/tan(x).
Plugging in the value we found for tan(x), cot(x) = 1/(√2/2).
Simplifying, cot(x) = 2/√2 = √2.
In summary:
sin(x) = -√(1/3)
cos(x) = -√2/3
tan(x) = √2/2
csc(x) = -√3
sec(x) = -√6/2
cot(x) = √2
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