cot (theta)=-9/8, cos (theta)<0
Are we solving for theta ??
if so, then
tan Ø = -8/9 and if cos Ø < 0
Ø must be in quadrant II
angle in standard position is 41.63°
( since tan 41.63° = +.888889 or 8/9 )
so Ø = 180 - 41.63 or 138.37°
check:
tan 138.37 = -.888889
cos 138.37 is negative.
Change your calculator to Radians if you want those units
To find the value of theta when cot(theta) = -9/8 and cos(theta) < 0, you can follow the steps below:
1. Recall the relationship between cotangent and cosine: cot(theta) = 1/tan(theta) = cos(theta)/sin(theta).
2. Since given cos(theta) is negative, we know that cos(theta) < 0.
3. Now, we have cot(theta) = -9/8 and cos(theta) < 0. We can use the relationship between cosine and cotangent to solve for theta.
4. Start by expressing cot(theta) in terms of cosine and sine: cot(theta) = cos(theta)/sin(theta).
5. Since cot(theta) = -9/8, we can substitute -9/8 for cos(theta)/sin(theta): -9/8 = cos(theta)/sin(theta).
6. Next, we can square both sides of the equation to eliminate the sin(theta): (-9/8)^2 = (cos(theta)/sin(theta))^2.
This simplifies to 81/64 = cos(theta)^2 / sin(theta)^2.
7. Now, use the Pythagorean Identity sin^2(theta) + cos^2(theta) = 1 to express cos(theta)^2 in terms of sin(theta)^2: sin(theta)^2 = 1 - cos(theta)^2.
8. Substitute sin(theta)^2 = 1 - cos(theta)^2 into equation (6): 81/64 = cos(theta)^2 / (1 - cos(theta)^2).
9. Rearrange equation (8) to isolate cos(theta)^2: cos(theta)^2 = (81/64) * (1 - cos(theta)^2).
10. Simplify equation (9): cos(theta)^2 = (81/64) - (81/64) * cos(theta)^2.
11. Combine like terms: cos(theta)^2 + (81/64) * cos(theta)^2 = 81/64.
12. Factor out cos(theta)^2: (1 + 81/64) * cos(theta)^2 = 81/64.
13. Combine the fractions: (64 + 81) / 64 * cos(theta)^2 = 81/64.
14. Simplify equation (13): (145/64) * cos(theta)^2 = 81/64.
15. Divide both sides by (145/64): cos(theta)^2 = (81/64) / (145/64).
16. Simplify equation (15): cos(theta)^2 = 81/145.
17. Take the square root of both sides to solve for cos(theta): cos(theta) = ±√(81/145).
18. Since cosine is negative (cos(theta) < 0) according to our given information, we can discard the positive square root and keep the negative one: cos(theta) = -√(81/145).
19. Simplify the expression: cos(theta) = -9/√145.
20. Given that cot(theta) = -9/8 and cos(theta) = -9/√145, we can substitute cos(theta) into the formula for cot(theta): -9/8 = (-9/√145) / sin(theta).
21. Rearrange the equation to solve for sin(theta): sin(theta) = (-9/√145) / (-9/8).
22. Simplify equation (21): sin(theta) = 8/√145.
23. Therefore, theta has a sine value of 8/√145 and cosine value of -9/√145.
24. To find the angle theta, use a scientific calculator by taking the inverse sine (sin^(-1)) or inverse cosine (cos^(-1)) of the values obtained for sine and cosine, respectively.
So, theta = sin^(-1)(8/√145) or theta = cos^(-1)(-9/√145).