Question

find the value of each trigonometric ratio.

1. tan c
2. tan x
3. tan x
4. sin a
5. tan c
6. cos x
7. cos c
8. sin x

Explanation:

Step1: Recall trigonometric - ratio formulas

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Solve for $\tan C$ in 1)

In the first right - triangle for $\tan C$, the opposite side to $\angle C$ is $48$ and the adjacent side is $14$. So, $\tan C=\frac{48}{14}=\frac{24}{7}$.

Step3: Solve for $\tan X$ in 2)

For $\tan X$, the opposite side to $\angle X$ is $35$ and the adjacent side is $12$. So, $\tan X = \frac{35}{12}$.

Step4: Solve for $\tan X$ in 3)

For $\tan X$, the opposite side to $\angle X$ is $6$ and the adjacent side is $8$. So, $\tan X=\frac{6}{8}=\frac{3}{4}$.

Step5: Solve for $\sin A$ in 4)

For $\sin A$, the opposite side to $\angle A$ is $6$ and the hypotenuse is $10$. So, $\sin A=\frac{6}{10}=\frac{3}{5}$.

Step6: Solve for $\tan C$ in 5)

For $\tan C$, the opposite side to $\angle C$ is $8$ and the adjacent side is $6$. So, $\tan C=\frac{8}{6}=\frac{4}{3}$.

Step7: Solve for $\cos X$ in 6)

For $\cos X$, the adjacent side to $\angle X$ is $24$ and the hypotenuse is $25$. So, $\cos X=\frac{24}{25}$.

Step8: Solve for $\cos C$ in 7)

For $\cos C$, the adjacent side to $\angle C$ is $40$ and the hypotenuse is $41$. So, $\cos C=\frac{40}{41}$.

Step9: Solve for $\sin X$ in 8)

For $\sin X$, the opposite side to $\angle X$ is $16$ and the hypotenuse is $20$. So, $\sin X=\frac{16}{20}=\frac{4}{5}$.

Answer:

1. $\tan C=\frac{24}{7}$
2. $\tan X=\frac{35}{12}$
3. $\tan X=\frac{3}{4}$
4. $\sin A=\frac{3}{5}$
5. $\tan C=\frac{4}{3}$
6. $\cos X=\frac{24}{25}$
7. $\cos C=\frac{40}{41}$
8. $\sin X=\frac{4}{5}$