Q47. - Algebra 2

Question

Explain why the sine and cosine of an acute angle are never greater than 1, but the tangent of an acute angle may be greater than 1.

Step-by-Step Solution

Answer

The sine and cosine of an acute angle are never greater than 1, but the tangent of an acute angle may be greater than 1.

Step 1. Given Information.

Given to explain why the sine and cosine of an acute angle are never greater than 1, but the tangent of an acute angle may be greater than 1.

Step 2. Explanation .

The sine function of an angle is defined as the ratio between the opposite side and hypotenuse of the angle of the right triangle:

$\sin \theta =\frac{\text{opp}}{\text{hyp}}$

 

The cosine function of an angle is defined as the ratio between the adjacent side and hypotenuse of the angle of the right triangle:

$\cos \theta =\frac{\text{adj}}{\text{hyp}}$

 

The tangent function of an angle is defined as the ratio between the opposite side and adjacent side of the angle of the right triangle:

$\tan \theta =\frac{\text{opp}}{\text{adj}}$

 

The longest side of a right triangle is hypotenuse, hence neither sine nor cosine can be greater than 1 whereas for right triangle when the opposite side is greater than the adjacent side, then the tangent for the angle can be greater than 1.

Step 3. Conclusion .

Hence, the sine and cosine of an acute angle are never greater than 1, but the tangent of an acute angle may be greater than 1.