Question

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. cosa
6. tana
7. sinc
8. cosc
9. tanc

(figure for 4 - 9: right triangle abc with right angle at b, ab = 14, bc = 48, ac = 50)

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

(figure for 10: right triangle abc with right angle at b, hypotenuse ac = 22, angle at c is 36°, side ab = x)

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

(figure for 11: right triangle abc with right angle at b, ab = 8, angle at a is 62°, hypotenuse ac = x)

Explanation:

First Section: Trig Ratios for Triangle ABC

Step1: Identify triangle sides

For right triangle $ABC$:

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

Step2: Calculate $\sin A$

$\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$
$\sin A = \frac{48}{50} = \frac{24}{25} = 0.96$

Step3: Calculate $\cos A$

$\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\cos A = \frac{14}{50} = \frac{7}{25} = 0.28$

Step4: Calculate $\tan A$

$\tan A = \frac{\text{opposite}}{\text{adjacent}}$
$\tan A = \frac{48}{14} = \frac{24}{7} \approx 3.43$

Step5: Calculate $\sin C$

$\sin C = \frac{\text{opposite}}{\text{hypotenuse}}$
$\sin C = \frac{14}{50} = \frac{7}{25} = 0.28$

Step6: Calculate $\cos C$

$\cos C = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\cos C = \frac{48}{50} = \frac{24}{25} = 0.96$

Step7: Calculate $\tan C$

$\tan C = \frac{\text{opposite}}{\text{adjacent}}$
$\tan C = \frac{14}{48} = \frac{7}{24} \approx 0.29$

Step1: Identify trigonometric ratio

$x$ is opposite $\angle C = 36^\circ$, hypotenuse = 22. Use $\sin\theta$.
$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$

Step2: Substitute values and solve

$x = 22 \times \sin(36^\circ)$
$x \approx 22 \times 0.5878 \approx 12.9$

Step1: Identify trigonometric ratio

$x$ is hypotenuse, side $AB=8$ is adjacent to $\angle A = 62^\circ$. Use $\cos\theta$.
$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

Step2: Rearrange and solve for $x$

$x = \frac{8}{\cos(62^\circ)}$
$x \approx \frac{8}{0.4695} \approx 17.0$

Answer:

Ratio	Simplified Fraction	Decimal
5. $\cos A$	$\frac{7}{25}$	$0.28$
6. $\tan A$	$\frac{24}{7}$	$3.43$
7. $\sin C$	$\frac{7}{25}$	$0.28$
8. $\cos C$	$\frac{24}{25}$	$0.96$
9. $\tan C$	$\frac{7}{24}$	$0.29$

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Question 10: Solve for $x$