Understanding the Expression sin90 - tan90 in Trigonometry

When dealing with trigonometric functions, it's important to understand the values and behaviors of these functions at specific angles. One common question in this domain is the evaluation of the expression sin90 - tan90. In this article, we will explore what this expression means and why it is undefined. We will also discuss related concepts like the values of sin90, cos90, and tan90.

The Behavior of Trigonometric Functions at Special Angles

Trigonometric functions, such as sine, cosine, and tangent, are defined based on the ratio of the sides of a right triangle or the unit circle. For an angle of 90 degrees (or u03C0/2 radians), these functions exhibit specific behaviors:

Sine of 90 Degrees

The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. At 90 degrees, the side opposite the angle has the maximum length, while the hypotenuse is the hypotenuse. Therefore:

sin90  1

Cosine of 90 Degrees

The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. At 90 degrees, the adjacent side becomes zero, making the cosine of 90 degrees equal to zero:

cos90  0

Tangent of 90 Degrees

The tangent of an angle is defined as the ratio of the sine to the cosine of that angle. For 90 degrees, this ratio becomes:

tan90  sin90 / cos90  1 / 0

Dividing by zero is undefined in mathematics, which means tan90 is undefined. Therefore, the expression sin90 - tan90 is also undefined because it involves using the undefined value of tan90.

Exploring the Expression sin90 - tan90

Given the values at 90 degrees, you might wonder if there are different ways to evaluate the expression sin90 - tan90. Here are a few attempts and their results:

Attempt 1: Direct Substitution

If you substitute the value of sin90 and proceed to evaluate tan90:

sin90 - tan90  1 - (sin90 / cos90)  1 - (1 / 0)

Since tan90 is undefined, the expression as a whole is undefined.

Attempt 2: Simplifying Before Substitution

Another approach might be to simplify the expression sin90 - sin90 / cos90 before substituting the values:

sin90 - sin90 / cos90

Substituting the values of sin90 and cos90 gives:

1 - 1 / 0

Again, since division by zero is undefined, this simplification does not lead to a defined value.

Attempt 3: Considering Limits

Another interesting approach is to consider the limits as the angle approaches 90 degrees:

lim(θ->90) sinθ - lim(θ->90) tanθ

Here, sinθ approaches 1, but tanθ approaches infinity as θ approaches 90 degrees. Therefore:

1 - lim(θ->90) sinθ / cosθ  1 - lim(θ->90) 1 / cosθ

As cosθ approaches 0, 1 / cosθ approaches infinity, so the result is:

1 - (∞)  -∞

This shows that the expression can approach negative infinity.

Summary

In conclusion, the expression sin90 - tan90 is undefined because tan90 is undefined due to the division by zero. While there are different ways to approach the problem, all lead to undefined or infinite results due to the inherent properties of trigonometric functions at 90 degrees.

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Frequently Asked Questions (FAQs)

FAQ 1: What is the value of sin90 and cos90?

The value of sin90 is 1, and the value of cos90 is 0.

FAQ 2: What is the value of tan90?

The value of tan90 is undefined because it involves division by zero.

FAQ 3: Is it possible to simplify the expression sin90 - tan90?

No, the expression is undefined because it involves the undefined value of tan90.