Question

find the value of each trigonometric ratio.

1. $\tan z$

1. $\cos c$

1. $\sin c$

1. $\tan x$

1. $\cos a$

1. $\sin a$

1. $\sin z$

1. $\sin c$

1. $\cos z$

1. $\tan c$

Explanation:

Step1: Recall trigonometric ratios

For a right triangle,

• $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
• $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

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1) Find $\tan Z$

Step1: Identify sides for $\angle Z$

Opposite = 21, Adjacent = 28

Step2: Compute $\tan Z$

$\tan Z = \frac{21}{28} = \frac{3}{4}$

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2) Find $\cos C$

Step1: Identify sides for $\angle C$

Adjacent = 16, Hypotenuse = 34

Step2: Simplify the ratio

$\cos C = \frac{16}{34} = \frac{8}{17}$

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3) Find $\sin C$

Step1: Identify sides for $\angle C$

Opposite = 28, Hypotenuse = 35

Step2: Simplify the ratio

$\sin C = \frac{28}{35} = \frac{4}{5}$

---

4) Find $\tan X$

Step1: Identify sides for $\angle X$

Opposite = 32, Adjacent = 24

Step2: Simplify the ratio

$\tan X = \frac{32}{24} = \frac{4}{3}$

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5) Find $\cos A$

Step1: Identify sides for $\angle A$

Adjacent = 30, Hypotenuse = 34

Step2: Simplify the ratio

$\cos A = \frac{30}{34} = \frac{15}{17}$

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6) Find $\sin A$

Step1: Identify sides for $\angle A$

Opposite = 32, Hypotenuse = 40

Step2: Simplify the ratio

$\sin A = \frac{32}{40} = \frac{4}{5}$

---

7) Find $\sin Z$

Step1: Identify sides for $\angle Z$

Opposite = 24, Hypotenuse = 40

Step2: Simplify the ratio

$\sin Z = \frac{24}{40} = \frac{3}{5}$

---

8) Find $\sin C$

Step1: Identify sides for $\angle C$

Opposite = 14, Hypotenuse = 50

Step2: Simplify the ratio

$\sin C = \frac{14}{50} = \frac{7}{25}$

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9) Find $\cos Z$

Step1: Identify sides for $\angle Z$

Adjacent = 24, Hypotenuse = 30

Step2: Simplify the ratio

$\cos Z = \frac{24}{30} = \frac{4}{5}$

---

10) Find $\tan C$

Step1: Identify sides for $\angle C$

Opposite = 36, Adjacent = 27

Step2: Simplify the ratio

$\tan C = \frac{36}{27} = \frac{4}{3}$

Answer:

1. $\boldsymbol{\frac{3}{4}}$
2. $\boldsymbol{\frac{8}{17}}$
3. $\boldsymbol{\frac{4}{5}}$
4. $\boldsymbol{\frac{4}{3}}$
5. $\boldsymbol{\frac{15}{17}}$
6. $\boldsymbol{\frac{4}{5}}$
7. $\boldsymbol{\frac{3}{5}}$
8. $\boldsymbol{\frac{7}{25}}$
9. $\boldsymbol{\frac{4}{5}}$
10. $\boldsymbol{\frac{4}{3}}$