Given: $$ \underline{\sin \alpha=\frac{3}{5} ext { and } 90^{\circ}

Question

Given: $$ \underline{\sin \alpha=\frac{3}{5} 	ext { and } 90^{\circ}<\alpha<270^{\circ}} $$ With the aid of a sketch and without using a calculator, determine: (a) $$ 	an \alpha $$ (b) $$ \cos \left(90^{\circ}+\alphaight) $$

UpStudy AI · Accepted Answer

Let's solve each part step by step using trigonometric identities and the given information.

### Given:
- $$ \sin \alpha = \frac{3}{5} $$
- $$ 90^{\circ} < \alpha < 270^{\circ} $$ (This places $$ \alpha $$ in the **second** or **third** quadrant.)

### (a) Determining $$ \tan \alpha $$

1. **Find $$ \cos \alpha $$:**

Using the Pythagorean identity:
   $$
   \sin^2 \alpha + \cos^2 \alpha = 1
   $$
   $$
   \left(\frac{3}{5}\right)^2 + \cos^2 \alpha = 1 \Rightarrow \frac{9}{25} + \cos^2 \alpha = 1
   $$
   $$
   \cos^2 \alpha = 1 - \frac{9}{25} = \frac{16}{25}
   $$
   $$
   \cos \alpha = \pm \frac{4}{5}
   $$
   
   **Determine the sign of $$ \cos \alpha $$:**
   
   - Since $$ \alpha $$ is in the **second** quadrant ($$ 90^{\circ} < \alpha < 180^{\circ} $$), where cosine is negative:
     $$
     \cos \alpha = -\frac{4}{5}
     $$

2. **Calculate $$ \tan \alpha $$:**
   $$
   \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4}
   $$

### **Answer for (a):**
$$
\tan \alpha = -\frac{3}{4}
$$

---

### (b) Determining $$ \cos \left(90^{\circ} + \alpha\right) $$

Use the cosine of a sum identity, particularly for angles involving $$ 90^{\circ} $$:

$$
\cos(90^{\circ} + \alpha) = -\sin \alpha
$$

Given $$ \sin \alpha = \frac{3}{5} $$:

$$
\cos(90^{\circ} + \alpha) = -\frac{3}{5}
$$

### **Answer for (b):**
$$
\cos \left(90^{\circ} + \alpha\right) = -\frac{3}{5}
$$

Answer:
(a) $$ \tan \alpha = -\dfrac{3}{4} $$;
(b) $$ \cos(90^\circ + \alpha) = -\,\dfrac{3}{5} $$
(a) $$ \tan \alpha = -\frac{3}{4} $$ (b) $$ \cos(90^\circ + \alpha) = -\frac{3}{5} $$

Given:
With the aid of a sketch and without using a calculator, determine:
(a)
(b)

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Solution

Given:

(a) Determining

Answer for (a):

(b) Determining

Answer for (b):

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