if csc= -7/10 and cos >0.
a) draw a triangle in the correct quadrant with three sides properly labled.
b) find the exact value of 7cot- 4cos. This means no decimals. find a common denominator and rationalize all denominators. simiplify completely
To draw the triangle in the correct quadrant, we need to use the given information: csc = -7/10 (cosecant is negative) and cos >0 (cosine is positive).
Let's start with the given information that csc = -7/10. The cosecant of an angle is defined as the reciprocal of the sine of that angle. In a right triangle, the sine of an angle is the length of the side opposite the angle divided by the length of the hypotenuse. So, we can write:
csc = 1/sin
Given that csc = -7/10, we can rewrite it as:
-7/10 = 1/sin
To find sin, we can take the reciprocal of -7/10:
sin = 10/-7 = -10/7
Since the sine is negative, we conclude that the angle is in the third or fourth quadrant.
Next, we are given that cos > 0. The cosine of an angle is defined as the length of the adjacent side divided by the length of the hypotenuse. Since cos > 0, we know that the adjacent side is positive.
Now, let's label the triangle with the given information. Since the angle is in the third or fourth quadrant, the sine will be negative. We can assume the hypotenuse is 10, and the opposite side is -10. Let's call the adjacent side "x" (a positive value):
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Now, let's move on to part b: finding the exact value of 7cot - 4cos. We need to find the values of cot (cotangent) and cos for the angle in the triangle we just drew.
The cotangent of an angle is defined as the reciprocal of the tangent of that angle. In a right triangle, the tangent of an angle is the length of the side opposite the angle divided by the length of the side adjacent to the angle.
Since we know the sine and cosine of the angle, we can find the tangent by dividing the opposite side by the adjacent side:
tan = sin/cos = (-10/7) / x = -10 / (7x)
To find cot, we can take the reciprocal of the tangent:
cot = 1/tan = 1 / (-10 / (7x)) = -7x/10
Now, let's substitute the values of cot and cos into the expression 7cot - 4cos:
7cot - 4cos = 7(-7x/10) - 4cos
Simplifying, we get:
7cot - 4cos = -49x/10 - 4cos
This is the exact value of 7cot - 4cos. We are not able to simplify it further without more information about the value of "x."