3) If $$ \mathrm{A}=20^{\circ} $$ and $$ \mathrm{B}=30^{\circ} $$ determine the value of the following correct to 1 decimal place: a) $$ 2 \sin B $$ e) $$ \cos \left(90^{\circ}-B\right) $$ b) $$ \frac{\sqrt{3}}{2} \cos B $$ f) $$ 2 \sin \left(2,5^{*}+A\right) \cos \left(2,5^{*}+A\right) $$ e) $$ \cos (A+2 B) $$ g) $$ \cos ^{2}(A)+\sin ^{2}(A) $$ d) $$ \sin \left(90^{\circ}-B\right) $$ h) $$ \left\{\cos ^{2}(B)-\sin ^{2}(B)\right. $$ Solntions 10 bivaces 1 umber 3 are on phye 293

Question

UpStudy AI · Accepted Answer

Sure, let's solve each part step-by-step using the given angles $$ A = 20^{\circ} $$ and $$ B = 30^{\circ} $$. All answers are rounded to one decimal place.

---

### a) $$ 2 \sin B $$

**Calculation:**
$$
2 \sin B = 2 \sin 30^{\circ} = 2 \times 0.5 = 1.0
$$

**Answer:**
$$
2 \sin B = 1.0
$$

---

### b) $$ \frac{\sqrt{3}}{2} \cos B $$

**Calculation:**
$$
\frac{\sqrt{3}}{2} \approx 0.8660
$$
$$
\cos 30^{\circ} \approx 0.8660
$$
$$
\frac{\sqrt{3}}{2} \cos B = 0.8660 \times 0.8660 \approx 0.75
$$

**Answer:**
$$
\frac{\sqrt{3}}{2} \cos B = 0.8 \quad \text{(rounded to 1 decimal place)}
$$

*Note:* For higher precision, $$ 0.8660 \times 0.8660 = 0.7500 $$, which is exactly $$ 0.75 $$. However, rounding to one decimal place gives **0.8**.

---

### c) $$ \cos \left(90^{\circ} - B\right) $$

**Calculation:**
$$
\cos(90^{\circ} - 30^{\circ}) = \cos 60^{\circ} = 0.5
$$

**Answer:**
$$
\cos \left(90^{\circ} - B\right) = 0.5
$$

---

### d) $$ \sin \left(90^{\circ} - B\right) $$

**Calculation:**
$$
\sin(90^{\circ} - 30^{\circ}) = \sin 60^{\circ} \approx 0.8660
$$
$$
\text{Rounded to 1 decimal place: } 0.9
$$

**Answer:**
$$
\sin \left(90^{\circ} - B\right) = 0.9
$$

---

### e) $$ \cos (A + 2B) $$

**Calculation:**
$$
A + 2B = 20^{\circ} + 2 \times 30^{\circ} = 80^{\circ}
$$
$$
\cos 80^{\circ} \approx 0.1736
$$
$$
\text{Rounded to 1 decimal place: } 0.2
$$

**Answer:**
$$
\cos (A + 2B) = 0.2
$$

---

### f) $$ 2 \sin \left(2.5^{\circ} + A\right) \cos \left(2.5^{\circ} + A\right) $$

**Calculation:**
$$
2 \sin C \cos C = \sin(2C) \quad \text{where } C = 2.5^{\circ} + 20^{\circ} = 22.5^{\circ}
$$
$$
2C = 45^{\circ}
$$
$$
\sin 45^{\circ} \approx 0.7071
$$
$$
\text{Rounded to 1 decimal place: } 0.7
$$

**Answer:**
$$
2 \sin \left(2.5^{\circ} + A\right) \cos \left(2.5^{\circ} + A\right) = 0.7
$$

---

### g) $$ \cos^{2}(A) + \sin^{2}(A) $$

**Calculation:**
This is a fundamental trigonometric identity.
$$
\cos^{2} \theta + \sin^{2} \theta = 1
$$
$$
\cos^{2}(20^{\circ}) + \sin^{2}(20^{\circ}) = 1
$$

**Answer:**
$$
\cos^{2}(A) + \sin^{2}(A) = 1.0
$$

---

### h) $$ \cos^{2}(B) - \sin^{2}(B) $$

**Calculation:**
This is the double-angle identity for cosine.
$$
\cos^{2} B - \sin^{2} B = \cos(2B)
$$
$$
2B = 60^{\circ}
$$
$$
\cos 60^{\circ} = 0.5
$$

**Answer:**
$$
\cos^{2}(B) - \sin^{2}(B) = 0.5
$$

---

### Summary of Answers

| Part | Expression                              | Value |
|------|-----------------------------------------|-------|
| a)   | $$ 2 \sin B $$                           | 1.0   |
| b)   | $$ \frac{\sqrt{3}}{2} \cos B $$         | 0.8   |
| c)   | $$ \cos (90^{\circ} - B) $$             | 0.5   |
| d)   | $$ \sin (90^{\circ} - B) $$             | 0.9   |
| e)   | $$ \cos (A + 2B) $$                      | 0.2   |
| f)   | $$ 2 \sin (2.5^{\circ} + A) \cos (2.5^{\circ} + A) $$ | 0.7   |
| g)   | $$ \cos^{2}(A) + \sin^{2}(A) $$         | 1.0   |
| h)   | $$ \cos^{2}(B) - \sin^{2}(B) $$         | 0.5   |

*All values are rounded to one decimal place.*
Here are the answers rounded to one decimal place: a) $$ 2 \sin B = 1.0 $$ b) $$ \frac{\sqrt{3}}{2} \cos B = 0.8 $$ c) $$ \cos (90^{\circ} - B) = 0.5 $$ d) $$ \sin (90^{\circ} - B) = 0.9 $$ e) $$ \cos (A + 2B) = 0.2 $$ f) $$ 2 \sin (2.5^{\circ} + A) \cos (2.5^{\circ} + A) = 0.7 $$ g) $$ \cos^{2}(A) + \sin^{2}(A) = 1.0 $$ h) $$ \cos^{2}(B) - \sin^{2}(B) = 0.5 $$

If and determine the value of the following correct to 1 decimal place:
a)
e)
b)
f)
e)
g)
d)
h)

Solntions 10 bivaces 1 umber 3 are on phye 293

Upstudy AI Solution

Answer

Solution

a)

b)

c)

d)

e)

f)

g)

h)

Summary of Answers

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If and determine the value of the following correct to 1 decimal place: a) e) b) f) e) g) d) h) Solntions 10 bivaces 1 umber 3 are on phye 293