# Solving Triangles

The contents of this online page:

- – the problems 76 - 77 are presented with examples of solutions and answers for the lesson "Solving triangles";
- – the questions, how to find the solution of a triangle having used sine and cosine of an angle, are considered in the tests 78 - 81;
- – the solutions, how to find the angle, the side of the triangle, are explained in the problems 82 - 85.

**Problem 76.**

Given:

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's 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 sine theorem

, we obtain the equality

*c=*≈ 11

Answer: B = 37°19'; C = 82°41'; *c ≈ *11

***

**Problem 77. **

Given:

the triangle ΔABC, the sides of the triangle

a=6,3

b=6,3

C = 54°

Find: A, B - the angles between the sides of the triangle,

the length of the side *c*

Solution:

Since *a=**b*=6,3, then the triangle ΔABC - is isosceles.

Then A =B = (180° - 54°): 2 = 63°

Using the sine theorem

, we obtain the equality

*c =* = ≈ 5,7

Answer: A =B = 63°; *c *≈ 5,7

***

## Solving triangles having used the sine and the cosine of the angle

**Problem 78.**

Given:

the triangle ΔABC

A = 60°

B = 40°

*c*=14

Find: C - the angle of the triangle, the lengths of the sides *a,**b*

Solution:

C = 180° - (40° + 60°) = 80°

Using the sine theorem

, we obtain the expression

*a *= ≈ 12

* * b* *= ≈ 9

Answer: C = 80°; *a *≈ 12; *b *≈ 9

***

**Problem 79.**

Given:

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 trigonometric tables of the book "The four-valued mathematical tables" or the calculation's results on the scientific calculator, we find the value of the angle B

B = 84°16'

Using the formula of the cosine theorem, we find the cosine of the angle C

Cos C = = =

= ≈ 0,7562785

Using trigonometric tables of the book "The four-valued mathematical tables" or 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') = 55°32'

Answer: A = 55°32' ; C = 40°52' ; B = 84°16'

***

**Problem 80.**

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°, then the triangle ΔABC - is isosceles.

Therefore, the two sides are equal AC=AB=b=c=4,5

Using the sine theorem

,

we find the side BC=a

*a *= ≈ 2,3

Answer: B = 75°; *a *≈ 2,3 ; c* *= 4,5

***

**Problem 81.**

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, then the triangle ΔABC - is isosceles, and therefore acute.

2) Using the formula of the cosine theorem

, 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 cosine theorem

, 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 solving the triangle having used the angle between the sides

**Problem 82.**

Given:

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 sine theorem

, we find the side AC

AC = (3 • 1) • 2 = 6 (m)

Using the sine theorem

, we find the side AB

AB = ≈ 3 (m)

Using the sine theorem

, we find the side BC

BC =≈ 4 (m)

Answer: AB ≈ 3 m, AC = 6 m, BC ≈ 4 m.

***

**Problem 83.**

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, then using the formula of the cosine theorem

Cos C =, we find the cosine of the angle C

Cos C = = ≈ 0,24

Using trigonometric tables of the book "The four-valued mathematical tables" or the calculation on the scientific calculator, we find the approximate value of the angle C

C ≈ 76°13'

Using the formula of the cosine theorem

Cos B =, we find the cosine of the angle B

Cos B = ==≈ 0,4857

Using trigonometric tables of the book "The four-valued mathematical tables" or 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°50'

Answer: A ≈ 42°50' ; B ≈ 60°57' ; C ≈ 76°13'

***

**Problem 84.**

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 sine theorem

, 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.

***

**Problem 85.**

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 cosine theorem

, we obtain

*a = *= ≈ 13,8

Using the formula of the cosine theorem

Cos C =, we find the cosine of the angle C

Cos C == ≈ 0,7457

Using trigonometric tables of the book "The four-valued mathematical tables" or 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°53'

Answer: a ≈ 13,8 ; B ≈ 88°53' ; C ≈ 41°47'

***