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kumpel [21]
3 years ago
15

Find the values of θ in the range 0≤θ≤360° which satisfy: 2 sin^2 θ - sinθ -1= 0

Mathematics
1 answer:
Sergio [31]3 years ago
7 0

Answer:

Step-by-step explanation:

Solving trig equations are just like solving "regular" equations. Let's get to it. First and foremost we are going to make a "u" substitution. You'll use that all the time in calculus, if you choose to go that route. Let

sin^2 \theta=u^2 and sinθ = u. Making the substitution, the equation becomes:

2u^2-u-1=0

That looks like something that can be factored, right? If you throw it into the quadratic formula you get the factors:

(u - 1)(2u + 1) = 0

By the Zero Product Property, either u - 1 = 0 or 2u + 1 = 0, so we will solve those, but not until after we back-substitute!

Putting sinθ back in for u:

sinθ - 1 = 0 so

sinθ = 1 and in the other equation:

2sinθ + 1 = 0 so

2sinθ = -1 and

sin\theta=-\frac{1}{2}

Get out the unit circle and look to where the sinθ has a value of 1. There's only one place in your interval, and it's at 90 degrees.

Now look to where the sinθ has a value of -1/2. There are 2 places within your interval, and those are at 210° and 330°. Now you're done!

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The perimeter of the dodecagon is 54 cm. by joining every other vertex of the dodecagon a hexagon is formed. if each side of the
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x = 8.69

Step-by-step explanation:

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54/12 = 4.5 cm

if we draw the lines to remove 6 vertices and form a hexagon, 6 triangles with 2 sides of 4.5 cm are formed.

we know that the angle of each vertex is 150 ° because it is a dodecagon

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10. Polygon WXYZ has vertices W(-1.5, 1.5),
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Answer:

See explanation

Step-by-step explanation:

Polygon WXYZ has vertices W(-1.5, 1.5), X(6, 1.5), Y(6, -4.5), and Z(-1.5, -4.5).

Find the slopes of all sides:

WX:\ \dfrac{1.5-1.5}{6-(-1.5)}=\dfrac{0}{7.5}=0 - horizontal line

XY:\ \dfrac{-4.5-1.5}{6-6}=\dfrac{-6}{0} - slope undefined, so the line is vertical

YZ:\ \dfrac{-4.5-(-4.5)}{-1.5-6}=\dfrac{0}{-7.5}=0 - horizontal line

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Vertical and horizontal lines are perpendicular, so all angles W, X, Y and Z are right angles. Hence, quadrilateral WXYZ is a rectangle

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