Questions: Find sin(α) and cos(β), tan(α) and cot(β), and sec(α) and csc(β) of the following figure.

Find sin(α) and cos(β), tan(α) and cot(β), and sec(α) and csc(β) of the following figure.

Transcript text: Find $\sin (\alpha)$ and $\cos (\beta), \tan (\alpha)$ and $\cot (\beta)$, and $\sec (\alpha)$ and $\csc (\beta)$ of the following figure.

Solution

Solution Steps

Step 1: Find the hypotenuse

We have a right triangle with legs of length 15 and 25. We can use the Pythagorean theorem to find the length of the hypotenuse.

$h^2 = 15^2 + 25^2$ $h^2 = 225 + 625$ $h^2 = 850$ $h = \sqrt{850} = 5\sqrt{34}$

Step 2: Calculate sin(α) and cos(β)

$\sin(\alpha) = \frac{opposite}{hypotenuse} = \frac{15}{5\sqrt{34}} = \frac{3}{\sqrt{34}} = \frac{3\sqrt{34}}{34}$

$\cos(\beta) = \frac{adjacent}{hypotenuse} = \frac{15}{5\sqrt{34}} = \frac{3}{\sqrt{34}} = \frac{3\sqrt{34}}{34}$

Step 3: Calculate tan(α) and cot(β)

$\tan(\alpha) = \frac{opposite}{adjacent} = \frac{15}{25} = \frac{3}{5}$

$\cot(\beta) = \frac{adjacent}{opposite} = \frac{15}{25} = \frac{3}{5}$

Step 4: Calculate sec(α) and csc(β)

$\sec(\alpha) = \frac{hypotenuse}{adjacent} = \frac{5\sqrt{34}}{25} = \frac{\sqrt{34}}{5}$

$\csc(\beta) = \frac{hypotenuse}{opposite} = \frac{5\sqrt{34}}{15} = \frac{\sqrt{34}}{3}$

Final Answer:

(a) sin(α) = $\frac{3\sqrt{34}}{34}$ and cos(β) = $\frac{3\sqrt{34}}{34}$

(b) tan(α) = $\frac{3}{5}$ and cot(β) = $\frac{3}{5}$

(c) sec(α) = $\frac{\sqrt{34}}{5}$ and csc(β) = $\frac{\sqrt{34}}{3}$

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The TTC has set a bus mechanical reliability goal of 3,900 bus miles. Bus mechanical reliability is measured specifically as the number of bus miles between mechanical road calls. Suppose a sample of 100 buses resulted in a sample mean of 3,950 bus miles and a sample standard deviation of 175 bus miles. Complete parts (a) and (b) below. a. Is there evidence that the population mean bus miles is more than 3,900 bus miles? (Use a 0.05 level of significance.) State the null and alternative hypotheses. H0: μ = T H1: μ ≠ T (Type integers) Find the test statistic for this hypothesis test. The test statistic t= (Round to two decimal places as needed.) The critical value for the test statistic is(are) . (Round to two decimal places as needed) Is there sufficient evidence to reject the null hypothesis using α=0.05 ? A. Do not reject the null hypothesis. There is insufficient evidence at the 0.05 level of significance that the population mean bus miles is less than 3,900 bus miles. B. Reject the null hypothesis. There is sufficient evidence at the 0.05 level of significance that the population mean bus miles is greater than 3,900 bus miles. C. Reject the null hypothesis. There is sufficient evidence at the 0.05 level of significance that the population mean bus miles is less than 3,900 bus miles. D. Do not reject the null hypothesis. There is insufficient evidence at the 0.05 level of significance that the population mean bus miles is greater than 3,900 bus miles. b. The p-value is . (Round to 3 decimal places as needed.) What does this p-value mean given the results of part (a)? A. The p-value is the probability of getting a sample mean of 3,950 bus miles or greater if the actual mean is 3,900 bus miles. B. The p-value is the probability that the actual mean is 3,950 bus miles or less. C. The p-value is the probability that the actual mean is 3,900 bus miles or greater given the sample mean is 3,950 bus miles.

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