Find the size of all three angles in a triangle that has
side lengths 10 m, 7m and 13 m. Round each angle to 1 decimal place

Step-by-step explanation:

One angle can be found using the Law of Cosines:

  a² = b² +c² -2bc·cos(A)

  cos(A) = (b² +c² -a²)/(2bc)

Then the angle opposite the shortest side has the cosine ...

  cos(A) = (10² +13² -7²)/(2·10·13) = 220/260 = 11/13

The measure of that angle is ...

  A = arccos(11/13) ≈ 32.2042°

The measure of the second-largest angle is then found using the Law of Sines:

  sin(B)/sin(A) = b/a

  sin(B) = (b/a)sin(A) = 10/7·sin(32.32042°) ≈ 0.761341

  B = arcsin(0.761341) ≈ 49.5826°

The remaining angle makes the sum of angles be 180°.

  C = 180° -49.5826° -32.2042° = 98.2132°

Rounded to 1 decimal place, the angles opposite the given sides are ...

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Comment on the solution method

The Law of Cosines finds angle values without the ambiguity that is sometimes present when using the law of sines. If it is used to find the largest angle first (opposite the longest side), then that ambiguity is resolved.

We found the smallest two angles first, so did not use the Law of Sines to find the largest angle. That is how we chose to resolve any possible ambiguity here.

the angles are

81.8°

65.99°

32.21°

Step-by-step explanation:

Since we know the three sides and no angle is known, we would apply the law of Cosines which is expressed as

a² = b² + c² - 2abCosA

Where a,b and c are the length of each side of the triangle and A is the angle corresponding to a.

13² = 10² + 7² - 2(10 × 7)CosA

169 = 100 + 49 - 2(70)CosA

169 = 149 - 140CosA

140CosA = 169 - 149

140CosA = 20

CosA = 20/140

CosA = 0.143

A = Cos^(- 1)0.143

A = 81.8° to the nearest tenth

To find the second angle, B, we would apply the Sine rule which is expressed as

a/SinA = b/SinB = c/SinC

Therefore,

13/Sin81.8 = 7/SinB

Cross multiplying, it becomes

13 × SinB = 7 × Sin 81.8

13SinB = 7 × 0.99 = 6.93

SinB = 6.93/13 = 0.533

B = Sin^-1(0.533)

B = 32.21°

The sum of angles in a triangle is 180°. Therefore, the third angle is

180 - ( 32.21 + 81.8)

= 65.99°

2.

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