Question

approximation to the nearest ten - thousandth.
12.
\triangle abc with right angle at c, bc = 8, ac = 15, ab = 17
\tsin a =
\tcos a =
\t\tan a =
13.
\triangle xyz with right angle at y, xy = 21, yz = 28, xz = 35
\tsin z =
\tcos z =
\t\tan z =
14.
\triangle abc with right angle at b, ab = 14, bc = 48, ac = 50
\tsin c =
\tcos c =
\t\tan c =
15.
\triangle xyz with right angle at y, xy = 30, yz = 40, xz = 50
\tsin x =
\tcos x =
\t\tan x =

Explanation:

Step1: Recall trigonometric ratios

For a right triangle, for an acute angle $\theta$:
$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

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Problem 12: Right $\triangle ABC$, right at $C$

Sides: $BC=8$, $AC=15$, $AB=17$

Step1: Identify sides for $\angle A$

Opposite to $\angle A$: $BC=8$, Adjacent: $AC=15$, Hypotenuse: $AB=17$

Step2: Calculate $\sin A$

$\sin A = \frac{8}{17} \approx 0.4706$

Step3: Calculate $\cos A$

$\cos A = \frac{15}{17} \approx 0.8824$

Step4: Calculate $\tan A$

$\tan A = \frac{8}{15} \approx 0.5333$

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Problem 13: Right $\triangle XYZ$, right at $Y$

Sides: $XY=35$, $YZ=28$, $XY=21$

Step1: Identify sides for $\angle Z$

Opposite to $\angle Z$: $XY=21$, Adjacent: $YZ=28$, Hypotenuse: $XZ=35$

Step2: Calculate $\sin Z$

$\sin Z = \frac{21}{35} = 0.6000$

Step3: Calculate $\cos Z$

$\cos Z = \frac{28}{35} = 0.8000$

Step4: Calculate $\tan Z$

$\tan Z = \frac{21}{28} = 0.7500$

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Problem 14: Right $\triangle ABC$, right at $B$

Sides: $AB=14$, $BC=48$, $AC=50$

Step1: Identify sides for $\angle C$

Opposite to $\angle C$: $AB=14$, Adjacent: $BC=48$, Hypotenuse: $AC=50$

Step2: Calculate $\sin C$

$\sin C = \frac{14}{50} = 0.2800$

Step3: Calculate $\cos C$

$\cos C = \frac{48}{50} = 0.9600$

Step4: Calculate $\tan C$

$\tan C = \frac{14}{48} \approx 0.2917$

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Problem 15: Right $\triangle XYZ$, right at $Y$

Sides: $XY=30$, $YZ=40$, $XZ=50$

Step1: Identify sides for $\angle X$

Opposite to $\angle X$: $YZ=40$, Adjacent: $XY=30$, Hypotenuse: $XZ=50$

Step2: Calculate $\sin X$

$\sin X = \frac{40}{50} = 0.8000$

Step3: Calculate $\cos X$

$\cos X = \frac{30}{50} = 0.6000$

Step4: Calculate $\tan X$

$\tan X = \frac{40}{30} \approx 1.3333$

Answer:

Problem 12:

$\sin A = 0.4706$
$\cos A = 0.8824$
$\tan A = 0.5333$

Problem 13:

$\sin Z = 0.6000$
$\cos Z = 0.8000$
$\tan Z = 0.7500$

Problem 14:

$\sin C = 0.2800$
$\cos C = 0.9600$
$\tan C = 0.2917$

Problem 15:

$\sin X = 0.8000$
$\cos X = 0.6000$
$\tan X = 1.3333$