Question

How do I use the cosine rule for these problems?

Answer 1

Answer:

see explanation

Step-by-step explanation:

Using the Cosine rule

a² = b² + c² - 2abcosA ← Rearranging for cosA gives

cosA = $\frac{b^2+c^2-a^2}{2ab}$

(a)

let a = 10, b = 7, c = 8 and A = α, then

cosα = $\frac{7^2+8^2-10^2}{2(7)(8)}$ = $\frac{49+64-100}{112}$ = $\frac{13}{112}$, thus

α = $cos^{-1}$( $\frac{13}{112}$ ) ≈ 83° ( to the nearest degree )

(b)

let the angle opposite side 7 be x

Using the Sine rule

$\frac{10}{sin83}$ = $\frac{7}{sinx}$ ( cross- multiply )

10sinx = 7sin83 ( divide both sides by 10 )

sinx = $\frac{7sin83}{10}$ , thus

x = $sin^{-1}$ ( $\frac{7sin83}{10}$ ) = 44°

The third angle can be found using the sum of angles in a triangle

third angle = 180° - (83 + 44)° = 180° - 127° = 53°

The 3 angles are 83°, 44°, 53°