Question

$$sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$ find the value of that represents each trig ratio. leave your answer as a simplified fraction and a decimal, rounded to the nearest hundredth.

ratio	simplified fraction	decimal
5. $cos a$
6. $\tan a$
7. $sin c$
8. $cos c$
9. $\tan c$

(right triangle with right angle at b, side ab = 12, side bc = 16, hypotenuse ac = 20)

1. solve for $x$. round your answer to the nearest tenth. show all work to receive full credit.

(right triangle with right angle, one angle 53°, adjacent side to 53° is 11, opposite side is $x$)

1. solve for $x$. round your answer to the nearest tenth. show all work to receive full credit.

(right triangle with right angle, one angle 31°, opposite side to 31° is 14, hypotenuse is $x$)

Explanation:

For Questions 4-9:

First, identify sides for each angle:

• For $\angle A$: opposite = $BC=16$, adjacent = $AB=12$, hypotenuse = $AC=20$
• For $\angle C$: opposite = $AB=12$, adjacent = $BC=16$, hypotenuse = $AC=20$

Step4: $\sin A$ (opp/hyp)

$\sin A = \frac{16}{20} = \frac{4}{5}$
Decimal: $\frac{4}{5}=0.80$

Step5: $\cos A$ (adj/hyp)

$\cos A = \frac{12}{20} = \frac{3}{5}$
Decimal: $\frac{3}{5}=0.60$

Step6: $\tan A$ (opp/adj)

$\tan A = \frac{16}{12} = \frac{4}{3}$
Decimal: $\frac{4}{3}\approx1.33$

Step7: $\sin C$ (opp/hyp)

$\sin C = \frac{12}{20} = \frac{3}{5}$
Decimal: $\frac{3}{5}=0.60$

Step8: $\cos C$ (adj/hyp)

$\cos C = \frac{16}{20} = \frac{4}{5}$
Decimal: $\frac{4}{5}=0.80$

Step9: $\tan C$ (opp/adj)

$\tan C = \frac{12}{16} = \frac{3}{4}$
Decimal: $\frac{3}{4}=0.75$

Step1: Identify trig ratio

$\tan(53^\circ) = \frac{x}{11}$ (opp/adj)

Step2: Solve for $x$

$x = 11 \times \tan(53^\circ)$
$\tan(53^\circ)\approx1.3270$, so $x\approx11\times1.3270=14.597$

Step3: Round to nearest tenth

$x\approx14.6$

Step1: Identify trig ratio

$\tan(31^\circ) = \frac{14}{x}$ (opp/adj)

Step2: Rearrange to solve for $x$

$x = \frac{14}{\tan(31^\circ)}$
$\tan(31^\circ)\approx0.6009$, so $x\approx\frac{14}{0.6009}\approx23.298$

Step3: Round to nearest tenth

$x\approx23.3$

Answer:

Ratio	Simplified Fraction	Decimal
$\cos A$	$\frac{3}{5}$	$0.60$
$\tan A$	$\frac{4}{3}$	$1.33$
$\sin C$	$\frac{3}{5}$	$0.60$
$\cos C$	$\frac{4}{5}$	$0.80$
$\tan C$	$\frac{3}{4}$	$0.75$

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For Question 10: