if cscè= -7/10 and cosè >0.
a) draw a triangle in the correct quadrant with three sides properly labled.
b) find the exact value of 7cotè- 4cosè. This means no decimals. find a common denominator and rationalize all denominators. simiplify completely
To draw a triangle in the correct quadrant, we will use the given information about the values of cscè and cosè.
Since cscè is negative (-7/10) and cosè is positive, we can determine that è lies in the second quadrant. This is because in the second quadrant, both sine and cosecant are negative, while cosine is positive.
Let's label the sides of the triangle accordingly:
1. Opposite side: Since cscè = -7/10, the opposite side will have a length of 7, and its direction will be in the positive y-axis.
2. Adjacent side: Since cosè > 0, the adjacent side will have a positive length, but we don't have its exact value.
3. Hypotenuse: Let's label this as "r" since we don't have its exact value.
From the triangle, we can use the Pythagorean theorem to find the remaining side length:
r^2 = (opposite side)^2 + (adjacent side)^2
r^2 = 7^2 + (adjacent side)^2
r^2 = 49 + (adjacent side)^2
To find the exact value of 7cotè - 4cosè, we need to express both cotangent and cosine in terms of sine and cosine.
cotè = 1/tanè = cosè/sinè
cosè = cosè (as given)
Now, let's substitute the values we have:
7cotè - 4cosè = 7(cosè/sinè) - 4cosè
To simplify this expression, we need to find a common denominator for the fractions:
7(cosè/sinè) - 4cosè = (7cosè - 4cosè(sinè))/(sinè)
Next, we can distribute the -4cosè to get:
(7cosè - 4cosè(sinè))/(sinè) = (7cosè - 4cosè(1/cscè))/(sinè)
Since cscè = -7/10, we can simplify further:
(7cosè - 4cosè(1/cscè))/(sinè) = (7cosè + 4cosè/(7/10))/(sinè)
Now, let's rationalize the denominator by multiplying the fraction by its conjugate:
(7cosè + 4cosè/(7/10))/(sinè) = (7cosè + 4cosè(10/7))/(sinè)
Simplifying the expression further, we get:
(7cosè + 40cosè)/(7sinè) = (47cosè)/(7sinè)
Therefore, the exact value of 7cotè - 4cosè is 47cosè/(7sinè).