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Chapter 10: Problem 61

Explain why the sine or cosine of an acute angle cannot be greater than or equal to 1 .

Short Answer

Expert verified
The sine or cosine of an acute angle can't be greater than 1 because these functions describe the x and y coordinates respectively of a point on the unit circle, whose radius is 1, limiting their maximum value to the radius of the circle. Since acute angles are in the first quadrant where sin and cos are positive, their values must lie within the interval (0,1).

Step by step solution

01

Definition of Acute Angle

An acute angle is any angle that measures more than 0 degrees but less than 90 degrees. Acute angles exist in the first quadrant of the unit circle in trigonometry.
02

Sin and Cos in Unit Circle

In the unit circle, any point (x,y) along the circle's edge can be represented as (cos(theta), sin(theta)), where theta is the angle formed with the x-axis. The range of the sine and cosine functions in the unit circle, where the radius is 1, is from -1 to +1. This is because \( sin \theta \) and \( cos \theta \) represent the y and x coordinate respectively of a point on the circle, and those coordinates don't exceed the radius of the circle, which is 1.
03

Conclusion

Since acute angles exist solely within the first quadrant of the unit circle, where sin and cos are positive, the sine or cosine of an acute angle can neither exceed nor be equal to 1. They must lie within the interval (0,1).

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