finding angles
find the value of the labelled angles.
(a)
(b) 5.9 cm
(c) 2 cm
(d) 20 cm 5 cm
find the value of the labelled angles.
(e)
(f) 9 m 5 m
(g) 12.3 cm 5.8 cm
(h) 0.7 m 0.5 m
(i) 12.5 cm 9.4 cm
(j) 16.5 m 15.1 m
find the missing angles.
(k) 10 cm

Question

Accepted Answer

### Part (a)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the opposite side to $ x $ is $ 6\space\text{cm} $ and the adjacent side is $ 2\space\text{cm} $. We use the tangent function, where $ \tan(x)=\frac{\text{opposite}}{\text{adjacent}} $
$ \tan(x)=\frac{6}{2} = 3 $
## Step 2: Find the angle
To find $ x $, we take the arctangent (inverse tangent) of 3. $ x=\arctan(3) $
Using a calculator, $ x\approx71.57^{\circ} $

### Part (b)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the adjacent side to $ x $ is $ 4.6\space\text{cm} $ and the opposite side is $ 5.9\space\text{cm} $. We use the tangent function $ \tan(x)=\frac{\text{opposite}}{\text{adjacent}} $
$ \tan(x)=\frac{5.9}{4.6}\approx1.2826 $
## Step 2: Find the angle
$ x = \arctan(1.2826) $
Using a calculator, $ x\approx52.0^{\circ} $

### Part (c)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the opposite side to $ x $ is $ 8\space\text{m} $ and the hypotenuse is $ 10\space\text{m} $. We use the sine function $ \sin(x)=\frac{\text{opposite}}{\text{hypotenuse}} $
$ \sin(x)=\frac{8}{10}=0.8 $
## Step 2: Find the angle
$ x=\arcsin(0.8) $
Using a calculator, $ x\approx53.13^{\circ} $

### Part (d)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the adjacent side to $ x $ is $ 5\space\text{cm} $ and the hypotenuse is $ 20\space\text{cm} $. We use the cosine function $ \cos(x)=\frac{\text{adjacent}}{\text{hypotenuse}} $
$ \cos(x)=\frac{5}{20}=0.25 $
## Step 2: Find the angle
$ x = \arccos(0.25) $
Using a calculator, $ x\approx75.52^{\circ} $

### Part (e)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the opposite side to $ x $ is $ 12\space\text{mm} $ and the adjacent side is $ 48\space\text{mm} $. We use the tangent function $ \tan(x)=\frac{\text{opposite}}{\text{adjacent}} $
$ \tan(x)=\frac{12}{48}=0.25 $
## Step 2: Find the angle
$ x=\arctan(0.25) $
Using a calculator, $ x\approx14.04^{\circ} $

### Part (f)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the opposite side to $ x $ is $ 5\space\text{m} $ and the adjacent side is $ 9\space\text{m} $. We use the tangent function $ \tan(x)=\frac{\text{opposite}}{\text{adjacent}} $
$ \tan(x)=\frac{5}{9}\approx0.5556 $
## Step 2: Find the angle
$ x=\arctan(0.5556) $
Using a calculator, $ x\approx29.05^{\circ} $

### Part (g)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the adjacent side to $ x $ is $ 5.8\space\text{cm} $ and the hypotenuse is $ 12.3\space\text{cm} $. We use the cosine function $ \cos(x)=\frac{\text{adjacent}}{\text{hypotenuse}} $
$ \cos(x)=\frac{5.8}{12.3}\approx0.4715 $
## Step 2: Find the angle
$ x=\arccos(0.4715) $
Using a calculator, $ x\approx61.9^{\circ} $

### Part (h)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the adjacent side to $ x $ is $ 0.5\space\text{m} $ and the hypotenuse is $ 0.7\space\text{m} $. We use the cosine function $ \cos(x)=\frac{\text{adjacent}}{\text{hypotenuse}} $
$ \cos(x)=\frac{0.5}{0.7}\approx0.7143 $
## Step 2: Find the angle
$ x=\arccos(0.7143) $
Using a calculator, $ x\approx44.4^{\circ} $

### Part (i)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the opposite side to $ x $ is $ 9.4\space\text{cm} $ and the hypotenuse is $ 12.5\space\text{cm} $. We use the sine function $ \sin(x)=\frac{\text{opposite}}{\text{hypotenuse}} $
$ \sin(x)=\frac{9.4}{12.5}=0.752 $
## Step 2: Find the angle
$ x=\arcsin(0.752) $
Using a calculator, $ x\approx48.8^{\circ} $

### Part (j)
# Explanation:
## Step 1: Identify the sides relative to angle $ x $
In the right - triangle, the adjacent side to $ x $ is $ 15.1\space\text{m} $ and the hypotenuse is $ 16.5\space\text{m} $. We use the cosine function $ \cos(x)=\frac{\text{adjacent}}{\text{hypotenuse}} $
$ \cos(x)=\frac{15.1}{16.5}\approx0.9152 $
## Step 2: Find the angle
$ x=\arccos(0.9152) $
Using a calculator, $ x\approx24.0^{\circ} $

### Part (a) Answer: $ \approx71.6^{\circ} $
### Part (b) Answer: $ \approx52.0^{\circ} $
### Part (c) Answer: $ \approx53.1^{\circ} $
### Part (d) Answer: $ \approx75.5^{\circ} $
### Part (e) Answer: $ \approx14.0^{\circ} $
### Part (f) Answer: $ \approx29.1^{\circ} $
### Part (g) Answer: $ \approx61.9^{\circ} $
### Part (h) Answer: $ \approx44.4^{\circ} $
### Part (i) Answer: $ \approx48.8^{\circ} $
### Part (j) Answer: $ \approx24.0^{\circ} $

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Identify the sides relative to angle \( x \)

Step 2: Find the angle

Part (b)

Step 1: Identify the sides relative to angle \( x \)

Step 2: Find the angle

Part (c)

Step 1: Identify the sides relative to angle \( x \)

Step 2: Find the angle

Part (d)

Answer:

Step 1: Identify the sides relative to angle \( x \)

Step 2: Find the angle

Part (a) Answer: \( \approx71.6^{\circ} \)

Part (b) Answer: \( \approx52.0^{\circ} \)

Part (c) Answer: \( \approx53.1^{\circ} \)

Part (d) Answer: \( \approx75.5^{\circ} \)

Part (e) Answer: \( \approx14.0^{\circ} \)

Part (f) Answer: \( \approx29.1^{\circ} \)

Part (g) Answer: \( \approx61.9^{\circ} \)

Part (h) Answer: \( \approx44.4^{\circ} \)

Part (i) Answer: \( \approx48.8^{\circ} \)

Part (j) Answer: \( \approx24.0^{\circ} \)