solving using the law of sines or cosines. a=75°,B=?,C=?,a=18cm,b=9cm,c=?
You have a and A, so use sines:
a/sinA = b/sinB
That gives you B. Now A+B+C=180
Having C, use sines to get c.
To solve this problem using the Law of Sines or Cosines, we need to identify the given angle and side measurements.
Given:
a = 75°
a = 18 cm
b = 9 cm
To find angle B using the Law of Sines, we can set up the following equation:
sin(A) / a = sin(B) / b
Substituting the given values:
sin(75°) / 18 = sin(B) / 9
Next, we can find the value of sin(B) by cross-multiplying and simplifying the equation:
sin(B) = (9 * sin(75°)) / 18
Using a scientific calculator, we can find the sine of 75°, which is approximately 0.9659. Substituting this value:
sin(B) = (9 * 0.9659) / 18
Now, solving for B, we take the arcsine of both sides:
B = arcsin((9 * 0.9659) / 18)
Using a scientific calculator again, we find arcsin((9 * 0.9659) / 18) ≈ 66.42°
Therefore, angle B is approximately 66.42°.
To find angle C, we know that the sum of all angles in a triangle is 180°:
C = 180° - 75° - 66.42°
Calculating, we find that C ≈ 38.58°.
To find side c, we can use the Law of Cosines, which states:
c^2 = a^2 + b^2 - 2ab * cos(C)
Substituting the given values:
c^2 = 18^2 + 9^2 - 2 * 18 * 9 * cos(38.58°)
Simplifying further:
c^2 = 324 + 81 - 2 * 18 * 9 * cos(38.58°)
Now, using a scientific calculator, we find cos(38.58°) ≈ 0.7911:
c^2 = 324 + 81 - 2 * 18 * 9 * 0.7911
c^2 = 405 - 32.13
c^2 ≈ 372.87
Finally, calculating the square root of both sides:
c ≈ √372.87
c ≈ 19.3 cm
Therefore, side c is approximately 19.3 cm.