Question

a given triangle has sides x cm, 8 cm and (x + 2 ) cm where X>6. if the cosine of the large angle is 2/7 calculate the value of X and the size of the smallest angle and correct to nearest degrees​

Answer 1

x = 15, the smallest angle is 17 degrees

Explanation:

I added a photo of my solution

Answer 2

• x = 7; 48°

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Given a triangle with sides x cm, 8 cm and (x + 2) cm.

Since x > 6, the longest side is (x + 2) as it is longer than x and longer than 8.

We know the largest angle is opposite to the longest side, hence cosine of the angle opposite to (x + 2) is 2/7.

Use the law of cosines to find the value of x:

• (x + 2)² = x² + 8² - 2*x*8*(2/7)
• x² + 4x + 4 = x² + 64 - (32/7)x
• 4x + (32/7)x = 60
• (60/7)x = 60
• x = 7

The sides are:

• 7 cm, 8 cm and 9 cm

Find the sine of the largest angle, let us call it A:

• sine² A = 1 - cos² A
• sin² A = 1 - (2/7)²
• sin² A = 45/49
• sin A = √45/7

Use the law of sines to find the smallest angle, let it be C:

• a /sin A = c/sin C
• 9/(√45/7) = 7/ sin C
• sin C = √45/9
• C = arcsin (√45/9)
• C = 48° (rounded to the nearest degree)