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yaroslaw [1]
2 years ago
15

Find all exact solutions on the interval [0, 2π). Look for opportunities to use trigonometric identities.

Mathematics
2 answers:
Nostrana [21]2 years ago
7 0

Answer:

x = π/2, 7π/6, or 11π/6

Step-by-step explanation:

sin² x − cos² x − sin x = 0

Use Pythagorean identity to write cosine in terms of sine.

sin² x − (1 − sin² x) − sin x = 0

sin² x − 1 + sin² x − sin x = 0

2 sin² x − sin x − 1 = 0

Factor:

(2 sin x + 1) (sin x − 1) = 0

Solve:

sin x = -1/2 or 1

x = π/2, 7π/6, or 11π/6

Ipatiy [6.2K]2 years ago
5 0

Answer:

pi/2, 7pi/6, 11pi/12

Step-by-step explanation:

sin²x - cos²x - sinx = 0

sin²x - (1 - sin²x) - sinx = 0

2sin²x - sinx - 1 = 0

2sin²x - 2sinx + sinx - 1 = 0

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

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

sinx = -0.5, 1

sinx = 1 x = pi/2

sinx = -0.5 x = pi+pi/6 = 7pi/6

x = 2pi-pi/6 = 11pi/6

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For a mean oil change time of 20.51 minutes there would be a​ 10% chance of being at or​ below

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

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For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

\mu = 21.3, \sigma = 3.9, n = 40, s = \frac{3.9}{\sqrt{40}} = 0.6166

Treating this as a random​ sample, at what mean​ oil-change time would there be a​ 10% chance of being at or​ below?

This is the 10th percentile, which is X when Z has a pvalue of 0.1. So it is X when Z = -1.28.

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

-1.28 = \frac{X - 21.3}{0.6166}

X - 21.3 = -1.28*0.6166

X = 20.51

For a mean oil change time of 20.51 minutes there would be a​ 10% chance of being at or​ below

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