Question

Consider the case of a triangle with sides 5, 12, and the angle between them 90°.

a. What is the easiest method to find the missing side?
b. What is the easiest method to find the missing angles?
c. Can you use the law of cosines to find the missing side? If so, perform the calculations. If not, show why not.
d. Can you use the law of cosines to find the missing angles? If so, perform the calculations. If not, show why
not.
e. Consider a triangle with sides a, ????, and the angle between them 90°. Use the law of cosines to prove a wellknown theorem. State the theorem.
f. Summarize what you have learned in parts (a) through (e).

Answer 1

a) Applying Pithagoras Theorem

b) Trigonometric relations

c) No we can not

d) No we can not

e) See step-by-step explanation

f) See step-by-step-explanation

Explanation:

a) The easiest method for calculating the missing side is applying Pithagoras Theorem

c² = a² + b²

c² = (5)² + (12)² ⇒ c² = 25 + 144 ⇒ c² = 169 ⇒ c = 13

b) The easiest method to find the missing angles, is to calculate the sin∠ and then look for arcsin fuction in tables.

sin ∠α = 5/13 sin ∠α = 0.3846 ⇒ arcsin (0.3846 ) α = 23⁰

Now we can either get the other angle subtracting 180 - 90 - 23 = 67⁰

Or calculating sinβ = 12 /13 sinβ = 0.9230 arcsin(0.9230) β = 67⁰

c) The law of cosines should not be applied to right triangles, in fact for instance in our particular case we have:

Question a)

c² = a² + b² - 2*a*b*cos90⁰ (law of cosines ) but cos 90⁰ = 0

then c² = a² + b² which is the expression for theorem of Pithagoras

In case you look for calculating the missing angles

c² = a² + b² - 2*5*13*cos α

169 = 25 + 144 - 2*5*13* cos α

As you can see 169 - 25 - 144 = 0

Then we can not apply law of cosine in right triangles, we shoul apply trigonometric relations and Pithagoras theorem, and as we saw you can get the expression of Pithagoras theorem from cosine law

c² = a² + b² - 2*a*b*cos∠90° cos ∠ 90° = 0

Then c² = a² + b²