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Magazine Chapter 8: Problem 2
Describe cot \(t\) for \(0
Short Answer
Expert verified
cot \( t \) for \( 0 < t < \frac{\text{\text{蟺}}}{2} \) is \( \frac{\text{adjacent}}{\text{opposite}} \).
Step by step solution
01
Understand the definition of cotangent
The cotangent of an angle is the reciprocal of the tangent of that angle. Mathematically, this is written as \(\text{cot}(t) = \frac{1}{\text{tan}(t)}\).
02
Recall the definition of tangent for a right triangle
In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. This means \(\text{tan}(t) = \frac{\text{opposite}}{\text{adjacent}}\).
03
Express cotangent in terms of the sides of the triangle
Since \(\text{cot}(t) = \frac{1}{\text{tan}(t)}\), and \(\text{tan}(t) = \frac{\text{opposite}}{\text{adjacent}}\), we find that \(\text{cot}(t) = \frac{\text{adjacent}}{\text{opposite}}\).
04
Identify the applicable range
For \(0 < t < \frac{\text{\text{蟺}}}{2}\), the angle \( t \) is in the first quadrant of the unit circle, where all sine, cosine, tangent, and cotangent values are positive.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Ratio
In trigonometry, a ratio is the relationship between the lengths of two sides of a right triangle. These ratios are fundamental as they help to define the basic trigonometric functions: sine, cosine, and tangent.
- The sine of an angle is the ratio of the opposite side to the hypotenuse, \(\text{sin}(t) = \frac{\text{opposite}}{\text{hypotenuse}}\).
- The cosine of an angle is the ratio of the adjacent side to the hypotenuse, \(\text{cos}(t) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
- The tangent of an angle is the ratio of the opposite side to the adjacent side, \(\text{tan}(t) = \frac{\text{opposite}}{\text{adjacent}}\).
Tangent
The tangent function relates an angle of a right triangle to the ratio of its opposite side to its adjacent side. This is given by \(\text{tan}(t) = \frac{\text{opposite}}{\text{adjacent}}\).
When you are given a right triangle, and you know the lengths of the sides opposite and adjacent to the angle, you can easily find the tangent of that angle.
When you are given a right triangle, and you know the lengths of the sides opposite and adjacent to the angle, you can easily find the tangent of that angle.
- Example: If the opposite side is 3 units long and the adjacent side is 4 units long, the tangent of the angle is \(\text{tan}(t) = \frac{3}{4}\).
Reciprocal Trigonometric Function
The cotangent of an angle is known as a reciprocal trigonometric function because it is the reciprocal of the tangent function.
This means that: \(\text{cot}(t) = \frac{1}{\text{tan}(t)}\).
This means that: \(\text{cot}(t) = \frac{1}{\text{tan}(t)}\).
- If \(\text{tan}(t) = \frac{\text{opposite}}{\text{adjacent}}\), then \(\text{cot}(t) = \frac{\text{adjacent}}{\text{opposite}}\).
- Reciprocal relationships exist for other trigonometric functions as well. For example, secant is the reciprocal of cosine, and cosecant is the reciprocal of sine. This is given by: \(\text{sec}(t) = \frac{1}{\text{cos}(t)}\) and \(\text{csc}(t) = \frac{1}{\text{sin}(t)}\).
First Quadrant
In the coordinate plane, the first quadrant is where both x and y coordinates are positive. This quadrant includes angles between 0 and \(\frac{\text{蟺}}{2}\) radians (or 0 to 90 degrees).
For trigonometric functions in the first quadrant:
For trigonometric functions in the first quadrant:
- Sine, cosine, and tangent values are all positive.
- Consequently, their reciprocal functions (cosecant, secant, and cotangent) are also positive.
