Explain why the sine and cosine of an acute angle are never greater that 1, but the tangent of a

step 1 Hi everyone. So today we need to explain why the sine and cosine of an acute angle are never gre

Explain why the sine and cosine of an acute angle are never...

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Key Concepts

Sine and Cosine as Ratios in Right Triangles

In the context of right triangles, sine and cosine are defined as the ratios of the lengths of the sides to the hypotenuse. Given that the hypotenuse is the longest side in a right triangle, these ratios are necessarily less than or equal to 1, which reinforces the bounded nature of sine and cosine regardless of the size of the acute angle.

Unit Circle and Geometry of Trigonometric Functions

The unit circle provides a geometric framework in which sine and cosine are defined as the y-coordinate and x-coordinate of a point on a circle of radius 1, respectively. This intrinsic definition confines their values between -1 and 1, since no coordinate of a point on this circle can exceed the radius, providing an immediate explanation for why sine and cosine cannot be greater than 1.

Tangent as a Ratio of Sine to Cosine

The tangent function is defined as the ratio between sine and cosine. Since neither sine nor cosine is restricted in the same way when taken as a ratio (especially when the cosine is small), the tangent of an acute angle can exceed 1. This occurs as the cosine value decreases, making the ratio (sine divided by cosine) larger, and in certain cases can lead to values greater than 1.

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