if tan theta= -13/12 where theta is in 2nd quadrant find remaing trigonometric ratios
you have a 5-12-13 triangle
sin > 0
cos < 0
if theta terminates in quadrant 2 and sin theta equals 12/13 find sec theta
To find the remaining trigonometric ratios (sine, cosine, and cotangent) when tan(theta) = -13/12, where theta is in the second quadrant, we can use the following steps:
1. Start by using the formula for tangent: tan(theta) = opposite/adjacent. In the second quadrant, the tangent is negative, which means the opposite side is negative, and the adjacent side is positive.
2. Since tan(theta) = -13/12, we can assign the opposite side as -13 and the adjacent side as 12.
3. To find the hypotenuse, we can use the Pythagorean theorem: hypotenuse^2 = opposite^2 + adjacent^2.
Substituting the values, we get: hypotenuse^2 = (-13)^2 + 12^2 = 169 + 144 = 313.
Taking the square root of both sides, we find the hypotenuse = √313.
Now that we have the values for the opposite side, adjacent side, and hypotenuse, we can find the remaining trigonometric ratios:
1. Sine(theta) = opposite/hypotenuse = -13/√313 (opposite divided by hypotenuse).
2. Cosine(theta) = adjacent/hypotenuse = 12/√313 (adjacent divided by hypotenuse).
3. Cotangent(theta) = 1/tan(theta) = 1/(-13/12) = -12/13 (reciprocal of tangent).
Therefore, the remaining trigonometric ratios are:
Sine(theta) = -13/√313
Cosine(theta) = 12/√313
Cotangent(theta) = -12/13