Q2

Given:

A=68

B= ?

C= ?

a= ?

b= 8

c=5

Determine B

69.91

74.98

76.44

77.39

To determine the value of \( B \) given the variables, we need to know the specific context or equation that relates these variables. Since no direct formula or context is provided, I'll assume it's a problem involving the Law of Cosines, commonly used in trigonometry for solving triangles.

The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]

From the given information:
- \( A = 68 \) (which we'll assume is the angle opposite side \( a \))
- \( B \) (which we'll find, and it's the angle opposite side \( b \))
- \( C \) (the angle opposite side \( c \))
- \( a \) (not given, so let's consider solving in another way)
- \( b = 8 \)
- \( c = 5 \)

Using the Law of Cosines to find angle \( B \):
\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \]

First, let's use the Law of Cosines to find \( a \) (since we have \( A = 68^\circ \)).

\[ a^2 = b^2 + c^2 - 2bc \cos(A) \]
\[ a^2 = 8^2 + 5^2 - 2 \cdot 8 \cdot 5 \cdot \cos(68^\circ) \]
\[ a^2 = 64 + 25 - 80 \cdot \cos(68^\circ) \]
\[ a^2 = 89 - 80 \cdot 0.3746 \]
\[ a^2 = 89 - 29.968 \]
\[ a^2 = 59.032 \]
\[ a \approx 7.68 \]

Now, using this to find \( \cos(B) \):

\[ \cos(B) = \frac{7.68^2 + 5^2 - 8^2}{2 \cdot 7.68 \cdot 5} \]
\[ \cos(B) = \frac{59.032 + 25 - 64}{2 \cdot 7.68 \cdot 5} \]
\[ \cos(B) = \frac{20.032}{76.8} \]
\[ \cos(B) \approx 0.2609 \]

Finally, finding \( B \):

\[ B \approx \cos^{-1}(0.2609) \]
\[ B \approx 75.14^\circ \]

Given the choices:

- 69.91
- 74.98
- 76.44
- 77.39

The closest value to \( 75.14^\circ \) is \( 74.98^\circ \).

Therefore, \( B = 74.98^\circ \).

Would you like more details or have any questions?

Here are 5 more questions you might ask:

1. How can I solve for angle \( C \) in a triangle?
2. Can you explain the Law of Sines?
3. How do you find the area of a triangle given all sides?
4. What are the trigonometric identities?
5. How do you solve quadratic equations using the quadratic formula?

**Tip:** Always double-check your angle measurements to ensure they add up to 180° in a triangle!

Learn how to determine angle B in a triangle using the Law of Cosines with given side lengths and an angle. This problem covers trigonometric concepts and includes step-by-step calculations for finding the angle B. Suitable for high school students studying trigonometry.

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