Question

geo - b - 9.4 -9.5 - trigonometric functions worksheet
in exercises 3-8, find sin d, sin e, cos d, and cos e. write each answer as a fraction and as a decimal rounded to four places.
3.
4.
5.
6.
7.
8.

Explanation:

For Problem 3:

Step1: Define trigonometric ratios

For right $\triangle DFE$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$

Step2: Calculate $\sin D$

Opposite to $D$ is 12, hypotenuse=15.
$\sin D = \frac{12}{15} = \frac{4}{5} = 0.8000$

Step3: Calculate $\sin E$

Opposite to $E$ is 9, hypotenuse=15.
$\sin E = \frac{9}{15} = \frac{3}{5} = 0.6000$

Step4: Calculate $\cos D$

Adjacent to $D$ is 9, hypotenuse=15.
$\cos D = \frac{9}{15} = \frac{3}{5} = 0.6000$

Step5: Calculate $\cos E$

Adjacent to $E$ is 12, hypotenuse=15.
$\cos E = \frac{12}{15} = \frac{4}{5} = 0.8000$

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For Problem 4:

Step1: Define trigonometric ratios

For right $\triangle EFD$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$

Step2: Calculate $\sin D$

Opposite to $D$ is 35, hypotenuse=37.
$\sin D = \frac{35}{37} \approx 0.9459$

Step3: Calculate $\sin E$

Opposite to $E$ is 12, hypotenuse=37.
$\sin E = \frac{12}{37} \approx 0.3243$

Step4: Calculate $\cos D$

Adjacent to $D$ is 12, hypotenuse=37.
$\cos D = \frac{12}{37} \approx 0.3243$

Step5: Calculate $\cos E$

Adjacent to $E$ is 35, hypotenuse=37.
$\cos E = \frac{35}{37} \approx 0.9459$

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For Problem 5:

Step1: Define trigonometric ratios

For right $\triangle DFE$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$

Step2: Calculate $\sin D$

Opposite to $D$ is 28, hypotenuse=53.
$\sin D = \frac{28}{53} \approx 0.5283$

Step3: Calculate $\sin E$

Opposite to $E$ is 45, hypotenuse=53.
$\sin E = \frac{45}{53} \approx 0.8491$

Step4: Calculate $\cos D$

Adjacent to $D$ is 45, hypotenuse=53.
$\cos D = \frac{45}{53} \approx 0.8491$

Step5: Calculate $\cos E$

Adjacent to $E$ is 28, hypotenuse=53.
$\cos E = \frac{28}{53} \approx 0.5283$

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For Problem 6:

Step1: Define trigonometric ratios

For right $\triangle EFD$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$

Step2: Calculate $\sin D$

Opposite to $D$ is 36, hypotenuse=45.
$\sin D = \frac{36}{45} = \frac{4}{5} = 0.8000$

Step3: Calculate $\sin E$

Opposite to $E$ is 27, hypotenuse=45.
$\sin E = \frac{27}{45} = \frac{3}{5} = 0.6000$

Step4: Calculate $\cos D$

Adjacent to $D$ is 27, hypotenuse=45.
$\cos D = \frac{27}{45} = \frac{3}{5} = 0.6000$

Step5: Calculate $\cos E$

Adjacent to $E$ is 36, hypotenuse=45.
$\cos E = \frac{36}{45} = \frac{4}{5} = 0.8000$

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For Problem 7:

Step1: Define trigonometric ratios

For right $\triangle DFE$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$

Step2: Calculate $\sin D$

Opposite to $D$ is $13\sqrt{3}$, hypotenuse=26.
$\sin D = \frac{13\sqrt{3}}{26} = \frac{\sqrt{3}}{2} \approx 0.8660$

Step3: Calculate $\sin E$

Opposite to $E$ is 13, hypotenuse=26.
$\sin E = \frac{13}{26} = \frac{1}{2} = 0.5000$

Step4: Calculate $\cos D$

Adjacent to $D$ is 13, hypotenuse=26.
$\cos D = \frac{13}{26} = \frac{1}{2} = 0.5000$

Step5: Calculate $\cos E$

Adjacent to $E$ is $13\sqrt{3}$, hypotenuse=26.
$\cos E = \frac{13\sqrt{3}}{26} = \frac{\sqrt{3}}{2} \approx 0.8660$

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For Problem 8:

Step1: Define trigonometric ratios

For right $\triangle DFE$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$

Step2: Calculate $\sin D$

Opposite to $D$ is 8, hypotenuse=17.
$\sin D = \frac{8}{17} \approx 0.4706$

Step3: Calculate $\sin E$

Opposite to $E$ is 15, hypotenuse=17.
$\sin E = \frac{15}{17} \approx 0.8824$

Step4: Calculate $\cos D$

Adjacent to $D$ is 15, hypotenuse=17.
$\cos D = \frac{15}{17} \approx 0.8824$

Step5: Calculate $\cos E$

Adjacent to $E$ is 8, hypotenuse=17.
$\cos E = \frac{8}{17} \approx 0.4706$

Answer:

Problem 3:

$\sin D = \frac{4}{5} = 0.8000$, $\sin E = \frac{3}{5} = 0.6000$, $\cos D = \frac{3}{5} = 0.6000$, $\cos E = \frac{4}{5} = 0.8000$

Problem 4:

$\sin D = \frac{35}{37} \approx 0.9459$, $\sin E = \frac{12}{37} \approx 0.3243$, $\cos D = \frac{12}{37} \approx 0.3243$, $\cos E = \frac{35}{37} \approx 0.9459$

Problem 5:

$\sin D = \frac{28}{53} \approx 0.5283$, $\sin E = \frac{45}{53} \approx 0.8491$, $\cos D = \frac{45}{53} \approx 0.8491$, $\cos E = \frac{28}{53} \approx 0.5283$

Problem 6:

$\sin D = \frac{4}{5} = 0.8000$, $\sin E = \frac{3}{5} = 0.6000$, $\cos D = \frac{3}{5} = 0.6000$, $\cos E = \frac{4}{5} = 0.8000$

Problem 7:

$\sin D = \frac{\sqrt{3}}{2} \approx 0.8660$, $\sin E = \frac{1}{2} = 0.5000$, $\cos D = \frac{1}{2} = 0.5000$, $\cos E = \frac{\sqrt{3}}{2} \approx 0.8660$

Problem 8:

$\sin D = \frac{8}{17} \approx 0.4706$, $\sin E = \frac{15}{17} \approx 0.8824$, $\cos D = \frac{15}{17} \approx 0.8824$, $\cos E = \frac{8}{17} \approx 0.4706$