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In and . Find all possible values of , to the nearest th of a degree.Answer:
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Q. In and . Find all possible values of , to the nearest th of a degree.Answer:
- Law of Sines Formula: To find the possible values of angle , we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant for all three sides and angles in the triangle. The formula is:where , , and are the lengths of the sides, and , , and are the opposite angles. In our case, we have:We can plug in the values we know to find :First, we calculate :Now we can solve for :
- Calculating sin(G): Now that we have the value of , we can find angle by taking the inverse sine (arcsin) of . However, since the sine function is positive in both the first and second quadrants, there are two possible angles for that have the same sine value. These angles are supplementary, meaning they add up to degrees. We will find the first angle, which is the acute angle: degrees (to the nearest tenth)
- Finding Angle G: To find the second possible value for angle G, we subtract the first value from degrees: degrees degrees degreesSo the second possible value for angle G is degrees (to the nearest tenth).
- Checking Validity: We must now check if the second possible value for angle is valid by ensuring that the sum of angles in a triangle is degrees. We already have angle as degrees, and we need to check if angle can be a positive acute angle when angle is degrees.Sum of angles in a triangle = degreesAngle = degrees - Angle - Angle Angle = degrees - degrees - degreesAngle = degreesSince angle is positive and less than degrees, the second value for angle is valid.
