REASONING How does the tangent of an acute angle in a right triangle change as the angle measure

REASONING How does the tangent of an acute angle in a right...

Key Concepts

Monotonic Increase of the Tangent Function

The tangent function is monotonically increasing on the interval from 0 to 90 degrees (or 0 to ?/2 radians). This means that as the acute angle increases, the value of the tangent function increases as well. In a right triangle, as the acute angle increases, the side opposite the angle becomes relatively longer compared to the adjacent side, leading to a larger ratio, and thus a greater value of the tangent.

Tangent Function in Trigonometry

The tangent function is defined as the ratio of the length of the side opposite an angle to the length of the side adjacent to that angle in a right triangle. It can also be expressed as the quotient of the sine function divided by the cosine function. This function is an important trigonometric ratio that helps in relating the angles of a triangle to the ratios of its side lengths.

Acute Angles in Right Triangles

An acute angle in a right triangle is any angle that measures less than 90 degrees. In the context of trigonometry, considering acute angles is essential because trigonometric functions like tangent are typically defined and behave in predictable ways over the interval from 0 to 90 degrees, which corresponds to the inherent geometry of right triangles.

Transcript

00:01 So for this problem here, our goal is trying to rationalize the tangent ratio.

00:10 The tangent ratio is the opposite over the adjacent for an angle of x.

00:18 Now, if i were to increase x, then what happens to the tangent ratio of x? does it increase? does it decrease? does it not change at all? and we want to justify or answer.

00:37 So one thing that you can do to try to maybe hypothesize what happens is to just take tangent of 0 and a tangent of 1 degree and 2 degrees and keep doing this as much as you can in here.

00:53 But there's a better way to gain intuition for it.

00:57 Now, if you were to do like tangent 0 degrees, 1 degree, and 2 degrees, and so forth, you'll see that the value of tangent of x, the tangent of x, also increases.

01:08 But let's try to understand why that's the case, because we have to justify our answer.

01:17 So let's just take a generic right triangle, any right triangle, for that matter.

01:27 And let's call this the angle x.

01:32 Now for the tangent ratio, we only care about the opposite and the adjacent.

01:37 I know the bottom side is the adjacent side because this long side here, which is opposite to the right angle, is the hypotenuse.

01:46 So let's put that as the hypothesis.

01:48 This is the opposite.

01:50 This is the adjacent side.

01:55 So we know that the tangent of x is equal to the opposite over the adjacent, just by definition.

02:03 Now, if we were to increase the measure of the angle of x, right? one thing that you may recall here is that if i increase the measure of the angle of x, then the length of o has to increase.

02:21 The side length has to increase here...

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