Find the area of the triangle with the given measurements: a=7, 𝛼 = 45°, 𝛾 = 60°.
(To the extent possible use standard angles to simplify your responses. When necessary express your answers using the functions sin, cos, sin−1, cos−1. If there is no solution enter IMPOSSIBLE.)
Using standard convention in writing the data
You have triangle ABC, with angle A = 45°, and a = 7
angle B = 60°, which would make angle C = 75°
I can find AB using the sine law:
AB/sin60 = 7/sin45
AB = 7sin60/sin45 = 7(√3/2) / (1/√2)
= (7√3/2)(√2/1) = 7√6 / 2
Now you have 2 sides and the contained angle, just like in the
previous problem I did for you.
Repeat those steps to find the area
To find the area of a triangle, we can use the formula:
Area = (1/2) * a * b * sin(𝛾)
Given that we have the lengths of two sides, a = 7 and 𝛾 = 60°, we still need the length of the third side (b) to calculate the area.
To find the length of side b, we can use the Law of Sines:
sin(𝛼) / a = sin(𝛾) / b
Rearranging the formula, we have:
b = a * (sin(𝛾) / sin(𝛼))
Substituting the given values:
b = 7 * (sin(60°) / sin(45°))
Using the values of sin(60°) and sin(45°) from a table or calculator (approximately 0.866 and 0.707, respectively):
b ≈ 7 * (0.866 / 0.707)
b ≈ 8.61
Now that we have the lengths of all three sides, we can calculate the area using the formula:
Area = (1/2) * a * b * sin(𝛾)
Area = (1/2) * 7 * 8.61 * sin(60°)
Again, using the value of sin(60°) (approximately 0.866):
Area ≈ (1/2) * 7 * 8.61 * 0.866
Area ≈ 25.13
Therefore, the area of the triangle with side lengths a = 7, 𝛼 = 45°, and 𝛾 = 60° is approximately 25.13.
To find the area of a triangle given the lengths of two sides and the included angle, we can use the formula:
Area = (1/2) * a * b * sin(𝛾)
In this case, we are given the lengths of sides a=7 and the included angle 𝛾=60°. However, we need the length of the other side, side b, to calculate the area.
To find the length of side b, we can use the Law of Sines. The Law of Sines states that for any triangle:
a/sin(𝛼) = b/sin(𝛽) = c/sin(𝛾)
Here, 𝛼 represents the angle opposite side a, 𝛽 represents the angle opposite side b, and 𝛾 represents the angle opposite side c. In our triangle, we have the lengths of sides a and 𝛾 but not the angles 𝛼 and 𝛽.
To find angle 𝛽, we can use the fact that the sum of the angles in a triangle is 180°. We can calculate angle 𝛽 using the formula:
𝛽 = 180° - 𝛼 - 𝛾
Substituting the given values 𝛼=45° and 𝛾=60° into the equation, we have:
𝛽 = 180° - 45° - 60° = 75°
Now, we have the lengths of sides a=7, b, and the angles 𝛼=45° and 𝛽=75°. We can use the Law of Sines to find the length of side b:
7/sin(45°) = b/sin(75°)
Solving this equation for b, we have:
b = (7 * sin(75°))/sin(45°)
Using trigonometric identities, we know that sin(75°) can be expressed as sin(45° + 30°). We can use the sum-to-product identity for sine to simplify the expression:
sin(75°) = sin(45° + 30°) = sin(45°) * cos(30°) + cos(45°) * sin(30°)
sin(45°) = √2/2
cos(30°) = √3/2
cos(45°) = √2/2
sin(30°) = 1/2
Substituting these values back into the expression for b, we get:
b = (7 * (sin(45°) * cos(30°) + cos(45°) * sin(30°))) / sin(45°)
Now, you can calculate the value of b using the calculator:
b ≈ 7.92
Now that we have the lengths of sides a=7 and b≈7.92, and the angle 𝛾=60°, we can substitute these values into the formula for the area of a triangle:
Area = (1/2) * a * b * sin(𝛾)
= (1/2) * 7 * 7.92 * sin(60°)
Using a calculator:
Area ≈ 20.5 square units
Therefore, the area of the triangle with measurements a=7, 𝛼=45°, and 𝛾=60° is approximately 20.5 square units.