How do the values of trigonometric functions change as the angle increases on the unit circle in a counterclockwise direction?
Question
How do the values of trigonometric functions change as the angle increases on the unit circle in a counterclockwise direction?
Solution
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The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. The x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle.
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As the angle increases in a counterclockwise direction from 0 to 90 degrees (or 0 to π/2 radians), the values of the sine function increase from 0 to 1, while the values of the cosine function decrease from 1 to 0.
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As the angle continues to increase from 90 to 180 degrees (or π/2 to π radians), the values of the sine function decrease from 1 to 0, while the values of the cosine function continue to decrease, now from 0 to -1.
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From 180 to 270 degrees (or π to 3π/2 radians), the values of the sine function continue to decrease, now from 0 to -1, while the values of the cosine function increase from -1 to 0.
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Finally, from 270 to 360 degrees (or 3π/2 to 2π radians), the values of the sine function increase from -1 to 0, while the values of the cosine function continue to increase, now from 0 to 1.
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This cycle then repeats for angles greater than 360 degrees (or 2π radians). The values of the trigonometric functions continue to oscillate between -1 and 1 as the angle increases.
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The tangent function, which is the ratio of the sine to the cosine, also changes as the angle increases. It starts at 0, increases to positive infinity, drops to negative infinity and increases back to 0 as the angle goes from 0 to 180 degrees (or 0 to π radians). This cycle also repeats for angles greater than 180 degrees (or π radians).
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The cotangent, secant, and cosecant functions, which are the reciprocals of the tangent, cosine, and sine functions respectively, also change as the angle increases, but their behavior is more complex.
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