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Explain why the graphs of y = csc x and y = -k sin x will never intersect for any positive value of k...?
1. The expression (secθ + cscθ) / (tanθ + 1) is to be the left hand side of an equation that is an identity. Which one of the following four expression can be used as the right hand side of the equation to compete the identity
a) secθ
b) tanθ
c) cosθ
d) cscθ
2. The expression (cosθ - sin^2θ) / (cosθ) * cscθ is to be the left hand side of an equation that is an identity. Which one of the following four expression can be used as the right hand side of the equation to compete the identity
a) secθ + cosθ
b) cosθ - 1
c) cscθ - tanθ
d) tan^2θ
3. Explain why the graphs of y = csc x and y = -k sin x will never intersect for any positive value of k
4. cos^-1(-.9397)
a) 160 degrees
b) 157 degrees
c) 162 degrees
d) -20 degrees
please help best answer gets 10 points
1 Answer
- ?Lv 61 decade agoFavorite Answer
I don't care about the best answer, but I want to help you do these problems on your own.
1. You can rewrite the numerator of the expression as 1/sinx + 1/cosx. The common denominator is the product sinx(cosx), so the numerator can be written as
(cosx + sinx)/(sinx)(cosx). The denominator (in terms of sine and cosine) is
sinx/cosx + 1, which can be written as (sinx +cosx)/cosx (write the "1" as cosx/cosx).
Division is the same as multiplying by the reciprocal, so the expression becomes
[(cosx + sinx)/(sinx)(cosx)][(cosx)/(sinx + cosx)] = 1/sinx = csc x.
3. Think of what you know of the graphs of sinx and csc x; they share the points
(pi/2, 1), (3pi/2, -1) on the interval 0<=x<2pi. The "-k" in front of the sinx would flip the graph of sinx around the x axis and stretch or shrink the graph (depending on the value of k). So on the interval 0 to pi, the graph of cscx would be positive for all values of x (actually, greater than or equal to one) and the graph of -ksinx would be negative for all positive values of k (actually, less than or equal to -k for all positive values of k).
4. This is simply a calculator problem; make sure your calculator is in degrees, then take the inverse cosine of (-.9397). As a help, you know that the cosine is negative in quadrants II and III, so you can eliminate choice d right away.
I'll get back to you on #2.