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 solve the problem, we will start by drawing a triangle in the correct quadrant where csc is negative and cos is positive.
a) To draw the triangle, we will use the values given: csc = -7/10 and cos > 0.
1. Since csc = 1/sin, we can find the value of sin by taking the reciprocal of csc: sin = 1/(-7/10) = -10/7.
2. Since sin is negative and cos is positive, we know that the angle is in the third quadrant.
3. To draw the triangle, we label the sides as follows:
- The opposite side is -10, since sin is opposite/hypotenuse.
- The adjacent side is positive, as cos is adjacent/hypotenuse.
The triangle will have a right angle in the third quadrant with the following sides labeled:
- The hypotenuse is always positive, so let's choose a value for it, such as 10.
- The opposite side is -10.
- The adjacent side is positive, let's say 7.
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10 \
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-10 ___________\ 7
-3
b) To find the exact value of 7cot - 4cos, we first need to determine the values of cot and cos in terms of the triangle we drew.
1. Cot is the reciprocal of tan, which is adjacent/opposite.
- Since the adjacent side is 7 and the opposite side is -10, cot = 7/(-10) = -7/10.
2. Cos is the adjacent/hypotenuse.
- Since the adjacent side is 7 and the hypotenuse is 10, cos = 7/10.
Now, substitute the values into the expression 7cot - 4cos:
7cot - 4cos = 7(-7/10) - 4(7/10)
= -49/10 - 28/10
= (-49 - 28)/10
= -77/10
So, the exact value of 7cot - 4cos, without any decimals, is -77/10.