draw an angle x in the 1st quadrant with sinx=3/5, and an angle y in the 2nd quadrant with cosy=-12/13, then determine the exact value of tan(x+y).
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in QI, cosx = 4/5
in QII, sinx = 5/13
now you can get tanx and tany, and use the formula for tan(x+y)
To draw an angle in a specific quadrant, we need to determine the reference angle for that angle and then find the coordinates of a point on the unit circle that corresponds to that angle.
1. Angle x in the 1st quadrant with sin(x) = 3/5:
- Start by drawing the positive x-axis (horizontal) and the positive y-axis (vertical) on a coordinate plane.
- The reference angle for angle x is the angle between the terminal side of the angle and the positive x-axis. Since sin(x) = 3/5, we can deduce that the opposite side of the triangle is 3 and the hypotenuse is 5.
- Use the Pythagorean theorem to find the adjacent side:
adjacent^2 = hypotenuse^2 - opposite^2
adjacent^2 = 5^2 - 3^2
adjacent^2 = 25 - 9
adjacent^2 = 16
adjacent = 4 or -4 (We choose the positive value since x is in the 1st quadrant)
- Therefore, the coordinates of the point on the unit circle for angle x are (4, 3). Draw a line from the origin to this point.
2. Angle y in the 2nd quadrant with cos(y) = -12/13:
- The reference angle for angle y is the angle between the terminal side of the angle and the positive x-axis. Since cos(y) = -12/13, we can deduce that the adjacent side of the triangle is -12, and the hypotenuse is 13 (negative because it lies in the 2nd quadrant).
- Using the Pythagorean theorem, we find the opposite side:
opposite^2 = hypotenuse^2 - adjacent^2
opposite^2 = 13^2 - (-12)^2
opposite^2 = 169 - 144
opposite^2 = 25
opposite = 5 or -5 (We choose the negative value since y is in the 2nd quadrant)
- The coordinates of the point on the unit circle for angle y are (-12/13, -5). Draw a line from the origin to this point.
Now that we have the angles x and y with their corresponding points on the unit circle, we can determine the exact value of tan(x + y).
The formula for tan(x + y) is:
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x) * tan(y))
To find the value of tan(x), we divide the y-coordinate of point x by the x-coordinate:
tan(x) = 3/4
To find the value of tan(y), we divide the y-coordinate of point y by the x-coordinate:
tan(y) = -5 / (-12/13) = 65/12
Now we substitute these values into the formula:
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x) * tan(y))
tan(x + y) = (3/4 + 65/12) / (1 - 3/4 * 65/12)
Simplifying the expression:
tan(x + y) = (9/12 + 65/12) / (12/12 - 195/48)
tan(x + y) = (74/12) / (48/48 - 195/48)
tan(x + y) = 74/12 / -147/48
tan(x + y) = (74/12) * (-48/147)
tan(x + y) = -888/1764
tan(x + y) = -2/4
In simplified form, the exact value of tan(x + y) is -1/2.