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Question
1. Simplify: 1 - cos^4x / sin^2x
2. Prove Identity sinx tanx + cosx = secx
3. Evaluate all 6 trig functions where the given point lies on the terminal side of an angle in standard position.
a.(-3, 4timessqrootof3)
b.(20, -10)
4. If sin(x) = -3/7 and pi < x < pi/2 (greater than or equal to), determine sin(2x), cos(2x) and tan(2x)
1. Simplify: 1 - cos^4x / sin^2x
It can be said that 1 - cos^4x = (1-cos²x)(1+cos²x) and that sin²x = 1 - cos²x.
This gives us (1-cos²x)(1+cos²x)/(1-cos²x) = 1 + cos²x.
2. Prove Identity sinx tanx + cosx = secx
It is known that tanx = sinx / cosx, so sinx tanx = sin²x/cosx.
If we take sin²x/cosx + cosx and factor out 1/cosx, we get (1/cosx)(sin²x + cos²x).
It is known that sin²x + cos²x = 1, so this reduces to 1/cosx.
As one of the trig identities, this is also secx.
3. Evaluate all 6 trig functions where the given point lies on the terminal side of an angle in standard position.
Make the triangle with the hypotenuse of a triangle c, the far side from the ange being a,
and the near side to the angle as b. The angle will be A.
This gives sinA = a/c, cosA = b/c, tanA = a/b, ctnA = b/a, secA = c/b, and cscA = c/a.
a.(-3, 4√3) - this point is in the 2nd quadrant where x is negative and y is positive.
b.(20, -10) - this point is in the 4th quadrant where x is positive and y is negative.
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