4b The length of the side $$ A B $$ is reduced by 1 cm . The length of the side $$ B C $$ is still 7 cm . Angle $$ A C B $$ is still $$ 90^{\circ} $$ Will the value of $$ \cos A B C $$ increase or decrease? You must give a reason for your answer.

Question

UpStudy AI · Accepted Answer

In the given right-angled triangle $$ ABC $$ with $$ \angle ACB = 90^\circ $$, let's analyze the effect of reducing the length of side $$ AB $$ by 1 cm while keeping side $$ BC $$ constant at 7 cm.

1. **Original Configuration:**
   - **Side $$ AB $$**: Hypotenuse of the right-angled triangle.
   - **Side $$ BC $$**: One of the legs, fixed at 7 cm.
   - **Angle $$ ABC $$**: We're interested in the cosine of this angle, $$ \cos(\angle ABC) $$.

2. **Cosine of Angle $$ ABC $$:**
   $$
   \cos(\angle ABC) = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{BC}{AB}
   $$
   Since $$ BC $$ remains constant at 7 cm, the value of $$ \cos(\angle ABC) $$ depends inversely on the length of $$ AB $$.

3. **Effect of Reducing $$ AB $$ by 1 cm:**
   - **Decrease in $$ AB $$**: When the hypotenuse $$ AB $$ decreases, the denominator in the cosine ratio $$ \frac{BC}{AB} $$ becomes smaller.
   - **Resulting Increase in Cosine Value**: As the denominator decreases while the numerator remains the same, the overall value of $$ \cos(\angle ABC) $$ increases.

**Conclusion:** Reducing the length of side $$ AB $$ by 1 cm while keeping $$ BC $$ constant causes $$ \cos(\angle ABC) $$ to increase. This is because the adjacent side $$ BC $$ remains unchanged while the hypotenuse $$ AB $$ becomes shorter, making the ratio $$ \frac{BC}{AB} $$ larger.

**Final Answer:**  
Its cosine increases because the adjacent side stays the same while the hypotenuse becomes shorter, making cos ABC larger.
Reducing the length of side $$ AB $$ makes $$ \cos(\angle ABC) $$ increase because the adjacent side remains the same while the hypotenuse becomes shorter, resulting in a larger cosine value.

4b The length of the side \( A B \) is reduced by 1 cm . The length of the side \( B C \) is still 7 cm . Angle \( A C B \) is still \( 90^{\circ} \) Will the value of \( \cos A B C \) increase or decrease? You must give a reason for your answer.

Real Tutor Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Trigonometry Questions