Question

1. select all true statements.

a. ( ac ) is ( sqrt{119} )
b. ( ac ) is 13 units
c. ( cos(\theta) = \frac{5}{12} )
d. ( sin(alpha) = \frac{12}{13} )
e. ( \theta = arctanleft( \frac{5}{12}
ight) )
(triangle with right angle at b, ab=5, bc=12, angles at a is ( alpha ), at c is ( \theta ))

Explanation:

Step1: Calculate length of AC

Use Pythagorean theorem: $AC = \sqrt{AB^2 + BC^2} = \sqrt{5^2 + 12^2} = \sqrt{25+144} = \sqrt{169} = 13$

Step2: Verify $\cos(\theta)$

For angle $\theta$, adjacent side $BC=12$, hypotenuse $AC=13$. So $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}$

Step3: Verify $\sin(\alpha)$

For angle $\alpha$, opposite side $BC=12$, hypotenuse $AC=13$. So $\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13}$

Step4: Verify $\theta = \arctan(\frac{5}{12})$

For angle $\theta$, opposite side $AB=5$, adjacent side $BC=12$. So $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}$, thus $\theta = \arctan(\frac{5}{12})$

Answer:

b. AC is 13 units
d. $\sin(\alpha) = \frac{12}{13}$
e. $\theta = \arctan(\frac{5}{12})$