Final answer:
The two possible angles for BAC when triangle ABC has sides AB=61mm, AC=70mm, and angle ACB=59 degrees are 23.6 degrees and 53.1 degrees, calculated using the Law of Sines.
Step-by-step explanation:
The student provided information about triangle ABC: Angle ACB is 59 degrees, side AB is 61mm, and side AC is 70mm. To calculate the two possible sizes for angle BAC, we make use of the Law of Sines. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of the triangle. Here is a step-by-step approach to solve for angle BAC:
- Firstly, use the Law of Sines to set up the equation: \( \frac{AB}{\sin(ACB)} = \frac{AC}{\sin(BAC)} \).
- Next, plug in the known values to get: \( \frac{61mm}{\sin(59^\circ)} = \frac{70mm}{\sin(BAC)} \).
- Solve for \( \sin(BAC) \) and calculate the possible values for angle BAC.
- Remember that since the sine function is positive in the first and second quadrants, there are two possible angles for BAC that are less than 180 degrees and have the same sine value.
- Use a calculator to first find the acute angle, and then use the supplementary angle to find the obtuse angle if applicable.
The two possible sizes for the angle BAC, calculated to three significant figures, are 23.6 degrees and 53.1 degrees.