Complete the work to determine the value of $a$. 1. Use the law of sines: $\frac{\sin (A)}{a}=\frac{\sin (B)}{b}$. 2. Substitute: $\frac{\sin (45^{\circ})}{a}=\frac{\sin (77^{\circ})}{8}$. 3. Cross multiply: $8 \sin (45^{\circ})=a \sin (77^{\circ})$. 4. Solve for $a$ and round to the nearest hundredth: $a \approx$ $\square$ .
Answer (2)
Divide both sides of the equation by sin ( 7 7 ∘ ) to isolate a : a = s i n ( 7 7 ∘ ) 8 s i n ( 4 5 ∘ ) .
Substitute the values of sin ( 4 5 ∘ ) ≈ 0.7071 and sin ( 7 7 ∘ ) ≈ 0.9744 into the equation.
Calculate the value of a : a ≈ 0.9744 8 × 0.7071 ≈ 5.8056 .
Round the value of a to the nearest hundredth: 5.81 .
Explanation
Isolate a We are given the equation 8 sin ( 4 5 ∘ ) = a sin ( 7 7 ∘ ) and we need to solve for a . To isolate a , we divide both sides of the equation by sin ( 7 7 ∘ ) : a = sin ( 7 7 ∘ ) 8 sin ( 4 5 ∘ )
Substitute values We know that sin ( 4 5 ∘ ) ≈ 0.7071 and sin ( 7 7 ∘ ) ≈ 0.9744 . Substituting these values into the equation, we get: a = 0.9744 8 × 0.7071 a = 0.9744 5.6568 a ≈ 5.8056
Round to nearest hundredth Rounding the value of a to the nearest hundredth, we get: a ≈ 5.81
Final Answer Therefore, the value of a rounded to the nearest hundredth is approximately 5.81.
Examples
The law of sines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. It's incredibly useful in fields like surveying and navigation. For instance, imagine you're a surveyor trying to determine the distance to an inaccessible point, like a tall tree on the other side of a river. By measuring the angles to the tree from two known points and the distance between those points, you can use the law of sines to calculate the unknown distance to the tree. This principle allows for accurate measurements without needing to physically reach the object, making it an indispensable tool in various real-world applications.
To find the value of a , we use the law of sines and calculate a ≈ 5.81 after isolating it and substituting the appropriate sine values. Finally, we round the result to the nearest hundredth.
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