Question

1. solve for a:

triangle with sides 12, 16, angle 73° at a
solve for a.

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides (12, 16) and the included angle (73°) given. To find the third side \( a \) (opposite the included angle? Wait, no, in triangle \( ABC \), side \( a \) is opposite angle \( A \)? Wait, actually, in standard notation, side \( a \) is opposite angle \( A \), side \( b \) opposite angle \( B \), side \( c \) opposite angle \( C \). Wait, here angle at \( A \) is 73°, sides: \( AC = 12 \), \( AB = 16 \), and we need to find \( BC = a \). So we can use the Law of Cosines. The Law of Cosines formula for a triangle with sides \( b \), \( c \) and included angle \( A \) is \( a^{2}=b^{2}+c^{2}-2bc\cos(A) \). Here, \( b = 12 \), \( c = 16 \), \( A = 73^\circ \).

Step2: Plug in the Values

Substitute \( b = 12 \), \( c = 16 \), \( \cos(73^\circ) \approx 0.2924 \) into the formula:

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Step3: Calculate the Multiplication and Subtraction

First, calculate \( 384\times0.2924 \approx 384\times0.2924 = 112.2816 \)
Then, \( a^{2}=400 - 112.2816 = 287.7184 \)

Step4: Take the Square Root

Take the square root of \( 287.7184 \) to find \( a \):
\( a=\sqrt{287.7184}\approx16.96 \) (rounded to two decimal places) or approximately 17.

Answer:

\( a \approx 17 \) (or more precisely \( \approx 16.97 \))