Question

9a)
in △ abc,
a = 24, b = 33, c = 26
find m ∠ a

Explanation:

Step1: Recall the Law of Cosines

To find an angle in a triangle when all three sides are known, we use the Law of Cosines. The formula for angle \( A \) is:

$$ \cos A = \frac{b^2 + c^2 - a^2}{2bc} $$

Here, \( a = 24 \), \( b = 33 \), and \( c = 26 \).

Step2: Substitute the values into the formula

First, calculate \( b^2 \), \( c^2 \), and \( a^2 \):

• \( b^2 = 33^2 = 1089 \)
• \( c^2 = 26^2 = 676 \)
• \( a^2 = 24^2 = 576 \)

Now substitute these into the formula for \( \cos A \):

$$ \cos A = \frac{1089 + 676 - 576}{2 \times 33 \times 26} $$

Step3: Simplify the numerator and the denominator

• Numerator: \( 1089 + 676 - 576 = 1089 + 100 = 1189 \) (Wait, actually \( 676 - 576 = 100 \), so \( 1089 + 100 = 1189 \))
• Denominator: \( 2 \times 33 \times 26 = 66 \times 26 = 1716 \)

So,

$$ \cos A = \frac{1189}{1716} \approx 0.693 $$

Step4: Find the angle \( A \)

To find \( A \), take the inverse cosine (arccos) of \( 0.693 \):

$$ A = \arccos(0.693) \approx 46.1^\circ $$

Answer:

The measure of angle \( A \) is approximately \( 46.1^\circ \) (or in degrees, rounded to a reasonable decimal place). If we use a calculator for more precise calculation:

First, recalculate the numerator correctly: \( 33^2 + 26^2 - 24^2 = 1089 + 676 - 576 = 1089 + 100 = 1189 \) (correct). Denominator: \( 2\times33\times26 = 1716 \). Then \( \cos A = 1189 / 1716 \approx 0.693 \). Then \( A = \arccos(0.693) \approx 46.1^\circ \). So the answer is approximately \( 46.1^\circ \) (or \( 46^\circ \) if rounded to the nearest degree).