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Yanka [14]
2 years ago
7

Find the value of the trig function indicated.

Mathematics
2 answers:
11Alexandr11 [23.1K]2 years ago
7 0

Answer:

Solution given;

In right angled triangle with respect to θ

perpendicular: opposite side: [P]=12

base:adjacent:[b]=8

hypotenuse [h]=?

cosθ=?

According to the Pythagoras law

h²=p²+b²

h²=12²+8²

h=\sqrt{208}

h=4\sqrt{13}

Now

we have

Cos θ=\frac{adjacent}{hypotenuse}

Cos θ=\frac{8}{4\sqrt{13}}

by rationalising it

Cos θ= \frac{2\sqrt{13}}{\sqrt{13}×\sqrt{13}}

Cos θ= \frac{2\sqrt{13}}{13}

is a required answer.

jekas [21]2 years ago
6 0

Answer:

\displaystyle cos\theta = \frac{2\sqrt{13}}{13}

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right  

Equality Properties

  • Multiplication Property of Equality
  • Division Property of Equality
  • Addition Property of Equality
  • Subtraction Property of Equality

<u>Trigonometry</u>

[Right Triangles Only] Pythagorean Theorem: a² + b² = c²

  • a is a leg
  • b is another leg
  • c is the hypotenuse

[Right Triangles Only] SOHCAHTOA

[Right Triangles Only] cosθ = adjacent over hypotenuse

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify variables</em>

<em>a</em> = 8

<em>b</em> = 12

<em>c</em>

<u>Step 2: Solve for </u><em><u>c</u></em>

  1. Substitute in variables [Pythagorean Theorem]:                                          8² + 12² = c²
  2. Evaluate exponents:                                                                                       64 + 144 = c²
  3. Add:                                                                                                                 208 = c²
  4. [Equality Property] Square root both sides:                                                  √208 = c
  5. Rewrite:                                                                                                           c = √208
  6. Simplify:                                                                                                           c = 4√13

<u>Step 3: Define Pt. 2</u>

<em>Identify variables</em>

Angle θ

Adjacent leg = 8

Hypotenuse = 4√13

<u>Step 4: Find</u>

  1. Substitute in variables [Cosine]:                                                                   \displaystyle cos\theta = \frac{8}{4\sqrt{13}}
  2. Rationalize:                                                                                                     \displaystyle cos\theta = \frac{2\sqrt{13}}{13}
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Step-by-step explanation:

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

Normal Probability Distribution:

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

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Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

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

The mean of a population is 74 and the standard deviation is 15.

This means that \mu = 74, \sigma = 15

Question a:

Sample of 36 means that n = 36, s = \frac{15}{\sqrt{36}} = 2.5

This probability is 1 subtracted by the pvalue of Z when X = 78. So

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

By the Central Limit Theorem

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

Z = \frac{78 - 74}{2.5}

Z = 1.6

Z = 1.6 has a pvalue of 0.9452

1 - 0.9452 = 0.0548

0.0548 = 5.48% probability of a random sample of size 36 yielding a sample mean of 78 or more.

Question b:

Sample of 150 means that n = 150, s = \frac{15}{\sqrt{150}} = 1.2247

This probability is the pvalue of Z when X = 77 subtracted by the pvalue of Z when X = 71. So

X = 77

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

Z = \frac{77 - 74}{1.2274}

Z = 2.45

Z = 2.45 has a pvalue of 0.9929

X = 71

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

Z = \frac{71 - 74}{1.2274}

Z = -2.45

Z = -2.45 has a pvalue of 0.0071

0.9929 - 0.0071 = 0.9858

0.9858 = 98.58% probability of a random sample of size 150 yielding a sample mean of between 71 and 77.

c. A random sample of size 219 yielding a sample mean of less than 74.2

Sample size of 219 means that n = 219, s = \frac{15}{\sqrt{219}} = 1.0136

This probability is the pvalue of Z when X = 74.2. So

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

Z = \frac{74.2 - 74}{1.0136}

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Z = 0.2 has a pvalue of 0.5793

0.5793 = 57.93% probability of a random sample of size 219 yielding a sample mean of less than 74.2

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3 years ago
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