Question

find the value of each trigonometric ratio to the nearest ten - thousandth.

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

Explanation:

11) $\cos Z$

Step1: Recall the cosine formula

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. For angle $Z$, the adjacent side to $Z$ is $12$ and the hypotenuse is $15$.

Step2: Calculate the cosine value

$\cos Z=\frac{12}{15}=0.8$

12) $\cos C$

Step1: Check the Pythagorean theorem

First, verify if the triangle is a right - triangle. $36^{2}+27^{2}=1296 + 729=2025$ and $45^{2}=2025$. So, it is a right - triangle with the right - angle at $B$.

Step2: Use the cosine formula

For angle $C$, the adjacent side is $27$ and the hypotenuse is $45$. Then $\cos C=\frac{27}{45}=0.6$

13) $\tan C$

Step1: Recall the tangent formula

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For angle $C$, the opposite side to $C$ is $40$ and the adjacent side is $30$.

Step2: Calculate the tangent value

$\tan C=\frac{40}{30}=\frac{4}{3}\approx1.3333$

14) $\tan A$

Step1: Check the Pythagorean theorem

$21^{2}+20^{2}=441+400 = 841$ and $29^{2}=841$. So, it is a right - triangle with the right - angle at $B$.

Step2: Use the tangent formula

For angle $A$, the opposite side is $21$ and the adjacent side is $20$. Then $\tan A=\frac{21}{20}=1.05$

15) $\tan C$

Step1: Recall the tangent formula

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For angle $C$, the opposite side to $C$ is $12$ and the adjacent side is $35$.

Step2: Calculate the tangent value

$\tan C=\frac{12}{35}\approx0.3429$

16) $\tan X$

Step1: Check the Pythagorean theorem

$30^{2}+40^{2}=900 + 1600=2500$ and $50^{2}=2500$. So, it is a right - triangle with the right - angle at $Y$.

Step2: Use the tangent formula

For angle $X$, the opposite side is $30$ and the adjacent side is $40$. Then $\tan X=\frac{30}{40}=0.75$

17) $\sin Z$

Step1: Recall the sine formula

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For angle $Z$, the opposite side to $Z$ is $12$ and the hypotenuse is $37$.

Step2: Calculate the sine value

$\sin Z=\frac{12}{37}\approx0.3243$

18) $\sin Z$

Step1: Check the Pythagorean theorem

$30^{2}+40^{2}=900+1600 = 2500$ and $50^{2}=2500$. So, it is a right - triangle with the right - angle at $Y$.

Step2: Use the sine formula

For angle $Z$, the opposite side is $30$ and the hypotenuse is $50$. Then $\sin Z=\frac{30}{50}=0.6$

Answer:

1. $0.8$
2. $0.6$
3. $\approx1.3333$
4. $1.05$
5. $\approx0.3429$
6. $0.75$
7. $\approx0.3243$
8. $0.6$