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
Answer: 20 hours
Step-by-step explanation:
Equation: 15x = 300
Isolate the variable.
Divide both sides by 15: ¹⁵ˣ⁄₁₅ = ³⁰⁰⁄₁₅
x = 20
Ivan needs to work 20 hours in order to afford the bicycle.
Perimeter is 24.
Length is 4 greater than the width.
Let's use 'x' for width
we can write this as x+4 + x+4 + x + x
x+4 being the width+4 and x being the width
There are 4 sides to a rectangle so you are adding 4 values
Two length and two widths
so we can then combine the two lengths to make 2x+8
and then combine the two widths to make 2x
Then put it together. 2x+2x+8= 4x+8
4x+8=24
Subtract 8 from both sides
4x=16
divide both sides by 4
x=4
The width is 4.
