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Chapter 4: Problem 92

Use words (not an equation) to describe one of the quotient identities.

Short Answer

Expert verified
The quotient identity \(\tan 胃 = \frac{\sin 胃}{\cos 胃}\) in words describes that the tangent of an angle is equal to the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle in a right-angle triangle. This ratio is geometrically as the measure of 'steepness', or gradient, of the line.

Step by step solution

01

Choose the identity

Choose the quotient identity that will be described. Here, the identity \(\tan 胃 = \frac{\sin 胃}{\cos 胃}\) is selected.
02

Explain the identity

The identity says that the tangent of an angle in a right-angle triangle is equal to the ratio of the side opposite to the angle (which is the length represented by the sine of the angle) to the side adjacent to the angle (which is the length represented by the cosine of the angle). That means, geometrically, tangent represents a ratio of lengths of two sides in a right-angled triangle where both sides are related to the angle 胃.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent as a Ratio in Triangles
The tangent of an angle in a right-angled triangle offers a meaningful way to understand the relationship between the angle's opposite and adjacent sides. Imagine a right-angled triangle with an angle \( \theta \) (theta).
Tangent helps us determine how these two sides compare:
  • The opposite side: The side directly opposite the angle \( \theta \).
  • The adjacent side: The side that helps form the angle \( \theta \), but is not the hypotenuse.
The quotient identity for tangent states that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), meaning tan \( \theta \) equals the length of the opposite side (http://sine) divided by the adjacent side (https://cosine).
This identity is crucial for solving problems involving right triangles, as it shows that tangent is merely a ratio, simplifying the complexity of angular relationships.
Understanding Sine Function
Sine is one of the primary trigonometric functions that measures an angle's relationship to the length of the triangle sides. Consider a right-angled triangle:
  • The opposite side: This is the side opposite the angle \( \theta \) - crucial when talking about the sine function.
  • The hypotenuse: The longest side of the triangle, opposite the right angle.
The sine of an angle \( \theta \), written as \( \sin \theta \), is the ratio of the length of the opposite side to the hypotenuse: \[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Sine tells us how high a point on a circle would be if the circle's radius equals the hypotenuse. It plays a vital role in various applications, like oscillations and waves, echoing its frequent utility in fields such as physics and engineering.
Cosine Function in Right Triangles
Cosine, similar to sine, is another foundational trigonometric function that aids in understanding triangular relationships. For a given angle \( \theta \) in a right-angled triangle, the function focuses on:
  • The adjacent side: This is the side next to the angle \( \theta \) but not the hypotenuse.
  • The hypotenuse: The side opposite the right angle.
The cosine of an angle \( \theta \) is denoted by \( \cos \theta \) and defined as the ratio of the adjacent side to the hypotenuse: \[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Cosine can be visualized as how far a point on a unit circle would extend horizontally. Its practical applications span from calculating distances in navigation to solving complex engineering problems, underscoring its importance in various scientific and engineering disciplines.

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Most popular questions from this chapter

What is an angle?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using radian measure, I can always find a positive angle less than \(2 \pi\) coterminal with a given angle by adding or subtracting \(2 \pi\)

Graph: \(f(x)=\frac{2}{3} x-2.\) (Section 1.4, Example 4).

Define the sine of \(t\).

If \(\theta=\frac{3}{2},\) is this angle larger or smaller than a right angle?
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