Question

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the sides of the parallelogram are 6 cm and 14 cm and the angle between them is 30 degree. what are the lengths of its diagonal?
using trigonometry..

Answer 1

Diagonals can be calculated by cosine law..

If diagonal opposite to 30° = d1

Then, by cosine law…

d1² = 14² + 6² - 2*14*6 * cos30°

=> d1² = 196 + 36 - 168 * √3/2

=> d1² = 232 - 168* 0.865

=> d1² = 232 - 145.68

=> d1² = √86.68

=> d1 = 9.3 ( approx) ……………..(1)

Now, angle adjacent to 30° in parallelogram = (180- 30) = 150°

So, again by cosine law..

If diagonal opposite to 150° = d2

d2² = 14² + 6² - 2*14*6 *cos 150°

=> d2² = 196 + 36 - 168* cos(90°+60°)

=> d2² = 232 - 168 *( - sin60° )

=> d2² = 232 - 168 * (-√3/2)

=> d2² = 232 - 168 * - 0.86

=> d2² = 232 + 144.48

=> d2² = 376.48

=> d2 = √376.48

=> d2 = 19.4 ( approx) ……………….(2)

So, smaller diagonal = 9.3 cm ( approx)

& greater diagonal = 19.4 cm ( approx)