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The triangle ΔABC,
the sides of the triangle a=10, b=7
the angle A = 60°
Solve a triangle: B, C - the angles between the triangle sides,
- the length of the side c
Solution:
It is known that the sine formula
, we obtain the expression
Sin B = = = = ≈ 0.6062
Using Sin B ≈ 0.6062 and the trigonometric table of the book "The four-valued mathematical tables" or the calculation on the smartphone/ mobile phone: function "scientific calculator" – option "DEG" – button "SIN-1" – enter "0.6062" – button ")" – on the screen "asin(0.6062)" – button "=" – on the screen "37.315"
37.315° = 37° + 0.315 • 60’ = 37°19’
B = 37°19’
Then C = 180° - (60° + 37°19’) = 82°41’
Using the law of sines
, we obtain the equality
c=≈ 11
Answer: B = 37°19’; C = 82°41’; c ≈ 11
***
Given:
the triangle ΔABC, the sides of the triangle
a=6.3
b=6.3
Find: A, B - the angles between the sides of the triangle,
the length of the side c
Solution:
Since a=b=6.3, we see that the triangle ΔABC - is isosceles.
Then A =B = (180° - 54°): 2 = 63°
Using the law of sines
, we obtain the equality
c = = ≈ 5.7
Answer: A =B = 63°; c ≈ 5.7
***
Given:
the triangle ΔABC
A = 60°
c=14
Find: C - the angle of the triangle, the lengths of the sides a,b
Solution:
C = 180° - (40° + 60°) = 80°
Using the law of sines
, we obtain the expression
a = ≈ 12
b = ≈ 9
Answer: C = 80°; a ≈ 12; b ≈ 9
***
the triangle ΔABC
BC=a=6
AC=b=7.3
AB=c=4.8
Find: A, B, C - the angles of the triangle
Solution:
It is known that the cosine formula
, so we find the cosine of the angle B
Cos B = = = = ≈ 0.0998263
Using the calculation’s results on the scientific calculator, we find the value of the angle B
B = 84°16’
Using the formula of the law of cosines, we find the cosine of the angle C
Cos C = = =
= ≈ 0.7562785
Using the calculation’s results on the scientific calculator, we find the value of the angle C
C = 40°52’
Then the angle A A =180° - (40°52’ + 84°16’) = 54°52’
Answer: A = 54°52’ ; C = 40°52’ ; B = 84°16’
***
Given:
the triangle ΔABC
A = 30°
C = 75°
b = 4.5
Find: the angle B, the sides of the triangle a,c
Solution:
B = 180° - (30° + 75°) = 75°
Since two angles in the triangle are equal B =C = 75°, we see that the triangle ΔABC - is isosceles.
Therefore, the two sides are equal AC=AB=b=c=4.5
Using the law of sines
,
we find the side BC=a
a = ≈ 2.3
Answer: B = 75°; a ≈ 2.3 ; c = 4.5
***
Given:
the triangle ΔABC, the lengths of its sides
1) a=5 , b=c=4 | 2) a=5 , b=9 , c=6 | 3) a=17 , b=15 , c=8 |
Find: whether the triangle is obtuse, right, acute
Solution:
1) Since b=c=4, we see that the triangle ΔABC - is isosceles, and therefore acute.
, we find the cosine of the angle A
Cos A = = =0
Then the angle A is A = 90°. Therefore, the triangle ΔABC - is right.
3) Using the formula of the law of cosines
, we find the cosine of the angle B
Cos B = == -< 0.
Since the cosine of the angle B is less than zero, hence the angle B is obtuse, and the triangle ΔABC is obtuse.
***
the triangle ΔABC, two angles and the side
A = 45°
C = 30°
AD = 3 m
Find: the length of all sides of the triangle ΔABC = ?
Solution:
Knowing the size of two angles in the triangle ΔABC, we find the third angle B = 180° - (30° + 45°) = 105°
We find the angle DAB and consider ΔADC
DAB = 180° - (90° + 45 + 30°) = 15°
DAC = 15° + 45° = 60°
Using the law of sines
, we find the side AC
AC = (3 • 1) • 2 = 6 (m)
Using the law of sines
, we find the side AB
AB = ≈ 3 (m)
Using the law of sines
, we find the side BC
BC =≈ 4 (m)
Answer: AB ≈ 3 m, AC = 6 m, BC ≈ 4 m.
***
Given:
the triangle ΔABC
three sides a = 14, b = 18,
c = 20
Find:
all the angles of the triangle ΔABC = ?
Solution:
Since the larger side lies against the larger angle, we see that using the formula of the law of cosines
Cos C =, we find the cosine of the angle C
Cos C = = ≈ 0.24
Using the calculation on the scientific calculator, we find the approximate value of the angle C
C ≈ 76°07’
Using the formula of the law of cosines
Cos B =, we find the cosine of the angle B
Cos B = ==≈ 0.4857
Using the calculation on the scientific calculator, we find the approximate value of the angle B
B ≈ 60.941 ≈ 60°57’
Therefore, A = 180° - (76°13’ + 60°57’) ≈ 42°56’
Answer: A ≈ 42°56’ ; B ≈ 60°57’ ; C ≈ 76°07’
***
Given:
the triangle ΔEKP, the side and two angles
EP = 0.75
P = 40°
K = 25°
Find: the side of the triangle PK = ?
Solution:
Using the law of sines
, we find the side PK
E = 180° - (40° + 25°) =115°
Sin 115° = Sin (180° - 65°) = Sin 65°
Then
PK = ≈ 1.61
Answer: PK ≈ 1.61.
***
Given:
the triangle ΔABC, two sides and the angle
b = 18, c = 12
A = 50°
Find: solve the triangle - determine the values of the side and two angles
(a, B, C ) = ?
Solution:
Using the formula of the law of cosines
, we obtain
a = = ≈ 13.8
Using the formula of the law of cosines
Cos C =, we find the cosine of the angle C
Cos C == ≈ 0.7457
Using the calculation on the scientific calculator, we find the approximate value of the angle C
C ≈ 41°47’
Therefore, B = 180° - (50° + 41°47’) ≈ 88°13’
Answer: a ≈ 13.8 ; B ≈ 88°13’ ; C ≈ 41°47’
***