8.9 Given that $$ 5 \cos \beta-3=0 $$ and $$ 0^{\circ}

Question

8.9 Given that $$ 5 \cos \beta-3=0 $$ and $$ 0^{\circ}<\beta<90^{\circ} $$. If $$ \alpha+\beta=90^{\circ} $$ and $$ 0^{\circ}<\alpha<90^{\circ} $$, calculate the value of $$ \cot \alpha $$.

UpStudy AI · Accepted Answer

1. From the equation $$ 5 \cos \beta - 3 = 0 $$, solve for $$ \cos \beta $$:
   $$
   5 \cos \beta = 3 \quad \Longrightarrow \quad \cos \beta = \frac{3}{5}.
   $$

2. Since $$ 0^{\circ} < \beta < 90^{\circ} $$, the sine value is positive. Use the Pythagorean identity to find $$ \sin \beta $$:
   $$
   \sin \beta = \sqrt{1 - \cos^2 \beta} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}.
   $$

3. We are given that $$ \alpha + \beta = 90^{\circ} $$. This implies:
   $$
   \alpha = 90^{\circ} - \beta.
   $$

4. Recognize the complementary angle identity:
   $$
   \cot \alpha = \tan \beta.
   $$

5. Calculate $$ \tan \beta $$ using the sine and cosine values:
   $$
   \tan \beta = \frac{\sin \beta}{\cos \beta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}.
   $$

6. Therefore, the value of $$ \cot \alpha $$ is:
   $$
   \cot \alpha = \frac{4}{3}.
   $$
$$ \cot \alpha = \frac{4}{3} $$

8.9 Given that and .
If and , calculate the value of .

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8.9 Given that and . If and , calculate the value of .