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3 years ago

If the angles are represented in degrees, find both angles: sin(x+7)=cos(4x+8)

Mathematics

1 answer:

garri49 [273]3 years ago

7 0

Answer:

$m\angle 1 = 22\degree$

$m\angle 2 = 68\degree$

Step-by-step explanation:

$\sin(x + 7) = \cos(4x + 8) \\ \sin(x + 7) = \sin \{90 - (4x + 8) \} \\ \{ \because \cos \theta = \sin(90 \degree - \theta) \}\\ \therefore \: (x + 7) = 90 - (4x + 8) \\ x + 7 + 4x + 8 = 90 \\ 5x = 90 - 15 \\ 5x = 75 \\ x = \frac{75}{5} \\ x = 15$

(x + 7)° = (15 + 7)° = 22°

(4x + 8)° = (4*15 + 8)° = (60 + 8)° = 68°

$m\angle 1 = 22\degree$

$m\angle 2 = 68\degree$

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We can find the rate at which the distance from the plane to the station is increasing by imaging the formation of a right triangle with the following dimensions:

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b: is the other side of the triangle = the distance traveled by the plane when it is 4 miles away from the station and an altitude of 3 miles

h: is the hypotenuse of the triangle = distance between the plane and the station = 4 miles

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$\frac{d}{dt}(a^{2}) + \frac{d}{dt}(b^{2}) = \frac{d}{dt}(h^{2})$

$2a\frac{da}{dt} + 2b\frac{db}{dt} = 2h\frac{dh}{dt}$

Since "da/dt" is constant (the altitude of the plane does not change with time), we have:

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And knowing that the plane is moving at a speed of 500 mi/h (db/dt):

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$\frac{dh}{dt} = \frac{\sqrt{7} mi*500 mi/h}{4 mi} = 331 mi/h$

Therefore, the rate at which the distance from the plane to the station is increasing is 331 miles per hour.

I hope it helps you!

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