Answer to Question #154258 in Trigonometry for Angelo

Question

Find the other angle and sides of the triangle:

1. Right triangle ABC with angle b=38° of side a=14 cm.
2. Obtuse triangle DEF with sides d=12, e=16.3 cm, f=13.5 cm

Answer 1

QUESTION 1

Step 1: We know from the the law of sines

(a / sin A) = (b / sin B) = (c / sin C)

Therefore:

(b / sin B) = (a / sin A)

(b / sin 38⁰)= (14 / sin 52⁰)

b = (sin 38⁰ x 14) / sin 52⁰

b = 10.94 cm

Step 2: From pythogras theorem we know:

A² + B² = C²

14² + 10.94² = C²

315.68 = C²

C = 17.77 cm

Step 3: The interor angles of a triangle add up to 180⁰

90⁰ + 38⁰ + A = 180⁰

A = 180⁰ – 128⁰

A = 52⁰

A = 14, B= 10.94, and C = 17.77

Angle <A = 52⁰ <B = 90⁰  <C = 38⁰ 

QUESTION 2

Step 1: Given the lengths of all three sides of any triangle, each angle can be calculated using the following equations:

e = 16.3 cm

f = 13.5 cm 

d = 12 cm

<D = arcCos [(e² + f² – d²) /(2ef)] ... (eq i)

<E = arcCos [(d² + f² – e²) /(2df)] ...(eq ii)

<F = arcCos [(d² + e² – d²) /(2de)] ...(eq iii)

Step 2: Substitute the values to the equations i, ii, and iii

<D = arcCos [(16.3² + 13.5² – 12²) / (2 x 16.3 x 13.5)]

<D = 46.32⁰

<E = arcCos [(13.5² + 12² – 16.3²) /(2 x 13.5 x 12)]

< E = 79.23⁰

<F = arcCos [(16.3² + 12² – 13.5² ) /(2 x 16.3 x 13.5)]

< F = 54.45⁰

Therefore:

Angles <D = 46.32⁰  <E= 79.23⁰, and <F = 54.45⁰

Side lengths e = 16.3 cm , f = 13.5 cm , and d = 12 cm