By definition of tangent,
tan(2<em>θ</em>) = sin(2<em>θ</em>) / cos(2<em>θ</em>)
Recall the double angle identities:
sin(2<em>θ</em>) = 2 sin(<em>θ</em>) cos(<em>θ</em>)
cos(2<em>θ</em>) = cos²(<em>θ</em>) - sin²(<em>θ</em>) = 2 cos²(<em>θ</em>) - 1
where the latter equality follows from the Pythagorean identity, cos²(<em>θ</em>) + sin²(<em>θ</em>) = 1. From this identity we can solve for the unknown value of sin(<em>θ</em>):
sin(<em>θ</em>) = ± √(1 - cos²(<em>θ</em>))
and the sign of sin(<em>θ</em>) is determined by the quadrant in which the angle terminates.
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We're given that <em>θ</em> belongs to the third quadrant, for which both sin(<em>θ</em>) and cos(<em>θ</em>) are negative. So if cos(<em>θ</em>) = -4/5, we get
sin(<em>θ</em>) = - √(1 - (-4/5)²) = -3/5
Then
tan(2<em>θ</em>) = sin(2<em>θ</em>) / cos(2<em>θ</em>)
tan(2<em>θ</em>) = (2 sin(<em>θ</em>) cos(<em>θ</em>)) / (2 cos²(<em>θ</em>) - 1)
tan(2<em>θ</em>) = (2 (-3/5) (-4/5)) / (2 (-4/5)² - 1)
tan(2<em>θ</em>) = 24/7
9514 1404 393
Answer:
1 < x < 29
Step-by-step explanation:
The triangle inequality requires the sum of the two shortest sides exceed the longest side.
<u>When x and 14 are the shortest</u>:
x + 14 > 15
x > 1
<u>When 14 and 15 are the shortest</u>:
14 +15 > x
29 > x
Then the requirement for the length of x is ...
1 < x < 29
_____
<em>Additional comment</em>
The length of the third side of a triangle can be between the difference and sum of the two given sides.
the answer to this question (-2,3)
Hi, first thing to do is divide 2.5 by 4 which is 0.625and you can either multiply 0.625 by three or subtract it from 2.5. So the answer comes out to be 1.875. You’re welcome and have a nice day
