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anemone
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In a triangle $ABC$ with side lengths $a,\,b$ and $c$, it's given that $17a^2+b^2+9c^2=2ab+24ac$.
Evaluate $\cos \angle B$.
Evaluate $\cos \angle B$.
anemone said:In a triangle $ABC$ with side lengths $a,\,b$ and $c$, it's given that $17a^2+b^2+9c^2=2ab+24ac$.
Evaluate $\cos \angle B$.
kaliprasad said:we have $a^2 - 2ab + b^2 + 16a^2- 24ac + 9c^2 = (a-b)^2 + (4a-3c)^2 = 0$
hence $ a = b $ and $4a= 3c=>\frac{c}{a} = \frac{4}{3}$
hence $\cos \angle B = \frac{a^2+c^2-b^2}{2ac} = \frac{c^2}{2ac} = \frac{c}{2a} =\frac{2}{3} $
Cosine is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse of a right triangle.
To find the value of cosine for a specific angle in a triangle, you can use a scientific calculator or refer to a cosine table. Alternatively, you can use the Pythagorean theorem to calculate the adjacent side and the hypotenuse, and then use the definition of cosine to find its value.
The range of values for cosine is between -1 and 1. This means that the maximum value of cosine is 1, which occurs when the angle is 0 degrees or 180 degrees, and the minimum value is -1, which occurs when the angle is 90 degrees.
The value of cosine changes as the angle increases or decreases based on the trigonometric ratios. When the angle is acute (less than 90 degrees), the value of cosine increases as the angle increases. When the angle is obtuse (greater than 90 degrees), the value of cosine decreases as the angle increases.
Cosine is related to the other trigonometric functions (sine, tangent, cosecant, secant, and cotangent) through their respective ratios. For example, the sine of an angle is equal to the cosine of its complementary angle, and the tangent of an angle is equal to the sine divided by the cosine of that angle. These relationships are known as trigonometric identities and are useful for solving various trigonometric equations.