Why cos(3pi/7)=-cos(4pi/7)?

Consider all 4 quadrants. In the 1st quadrant, every trigonometric function is positive (sin, cos tan and their reciprocals). In the 2nd quadrant, only sin is positive. In the 3rd, only tangent is positive. In the 4th, only cosine is positive.

As a basic rule of thumb, a reference angle in quadrant 1 can be converted to another angle in another quadrant. For example, if we had a 30 degree angle, we can convert it to the second quadrant by doing 180 - 30 (or pi - pi/6), giving us 150 degrees or 5pi/6. If we take the sin, cosine, or tangent of 30, we will get positive numbers. If we take the sin of 150, we will get a positive equivalent number to sin 30 BECAUSE the sin is POSITIVE in the 2nd quadrant. If we take the cos or tan of 150, we will get NEGATIVE equivalent numbers to cos or tan of 30, respectively, because cos and tan are NEGATIVE in the 2nd quadrant.

3pi/7 is in the 1st quadrant, while 4pi/7 is in the second quadrant. We convert 3pi/7 to the second quadrant by subtracting it from pi. Naturally, pi- 3pi/7 is 4pi/7. While the cosine of 3pi/7 and 4pi/7 will be the same numerically, they will have different signs (cos 3pi/7 is positive, cos 4pi/7 is negative). If we negate cos 4pi/7 we will negate the negative value and have an answer equivalent to cos 3pi/7.

The angles are symmetric with respect to the x-axis, so their cosine are equal. Plot them on a unit circle.

Because the curve of cos(x) is symmetrical around x=pi/2.

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