/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Trigonometric identities Prove... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 30

Trigonometric identities Prove that \(\tan \theta=\frac{\sin \theta}{\cos \theta}\)

Short Answer

Expert verified
Question: Prove the trigonometric identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) using the definitions of \(\sin \theta\) and \(\cos \theta\) in terms of the sides of a right triangle. Answer: We have proved that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) by defining the sine and cosine of angle \(\theta\) using the sides of a right triangle, where the sine of the angle is the ratio of the side opposite the angle to the hypotenuse, and the cosine of the angle is the ratio of the side adjacent to the angle to the hypotenuse. We then expressed \(\tan \theta\) in terms of the sine and cosine, and simplified the expression.

Step by step solution

01

Understanding the trigonometric functions

The three main trigonometric functions, sine, cosine, and tangent, are defined based on a right triangle. If we have a right triangle with angle \(\theta\), the sine of the angle is the ratio of the length of the side opposite the angle to the length of the triangle's hypotenuse, and the cosine of the angle is the ratio of the length of the side adjacent to the angle to the hypotenuse. Using these definitions, we can find the tangent of an angle as well.
02

Define \(\tan \theta\) using \(\sin \theta\) and \(\cos \theta\)

To prove that \(\tan \theta=\frac{\sin \theta}{\cos \theta}\), we will start by defining the tangent of an angle. By definition, the tangent of an angle is the ratio of the sine of the angle to the cosine of the angle: \begin{equation} \tan \theta = \frac{\sin \theta}{\cos \theta} \end{equation}
03

Relate \(\sin \theta\) and \(\cos \theta\) using a right triangle

Let's consider a right triangle where \(\theta\) is one of the acute angles. We can label the sides of the triangle as: - \(a\): the side opposite angle \(\theta\) - \(b\): the side adjacent to angle \(\theta\) - \(c\): the hypotenuse (the side opposite the right angle) Using this triangle, we can define the sine and cosine of angle \(\theta\): \begin{align} \sin \theta &= \frac{a}{c} \\ \cos \theta &= \frac{b}{c} \end{align}
04

Express \(\tan \theta\) in terms of \(a\), \(b\), and \(c\)

Now, we can substitute the expressions for \(\sin \theta\) and \(\cos \theta\) from step 3 into the definition of \(\tan \theta\) from step 2: \begin{align} \tan \theta &= \frac{\sin \theta}{\cos \theta} \\ &= \frac{\frac{a}{c}}{\frac{b}{c}} \end{align}
05

Simplify the expression for \(\tan \theta\)

Finally, we will simplify the expression for \(\tan \theta\) by removing the fractions in the numerator and the denominator: \begin{align} \tan \theta &= \frac{\frac{a}{c}}{\frac{b}{c}} \\ &= \frac{a \cdot c}{b \cdot c} \\ &= \frac{a}{b} \end{align} Now, we have proved that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) using the definitions of \(\sin \theta\) and \(\cos \theta\) in terms of the sides of a right triangle.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Right Triangle
A right triangle is fundamental in understanding trigonometric functions. It is a triangle with one of its angles measuring 90 degrees. This right angle gives the triangle its name.
The remaining two angles are known as acute angles, each measuring less than 90 degrees.
  • The side opposite the right angle is the hypotenuse - the longest side of the triangle.
  • The other two sides are referred to as the opposite and adjacent sides with respect to an acute angle, \(\theta\).
The right triangle is crucial because it provides a geometric framework for defining the sine, cosine, and tangent functions. When you learn about these trigonometric functions, it is usually in the context of right triangles. Right triangles help in visualizing and understanding how these functions arise from basic geometric properties.
Sine Function
The sine function is one of the primary trigonometric functions, often denoted as \(\sin \theta\). It relates to the angles and sides of a right triangle. Specifically, it defines the ratio of the length of the side opposite an angle to the hypotenuse of the triangle. For a given angle \(\theta\), the sine function is represented as:\[\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\]This definition helps determine the size of an angle based on the triangle's dimensions or, conversely, allows for calculating a side length if the angle and length of the hypotenuse are known.
Key Points:
  • The sine value ranges between -1 and 1, inclusive, as it is a ratio comparing two sides of a triangle.
  • In right triangles, the sine function is useful for calculations involving the elevation or height relative to a baseline.
Cosine Function
The cosine function, or \(\cos \theta\), is another fundamental trigonometric function. It is defined in terms of a right triangle similar to the sine function. The cosine of an angle \(\theta\) is the ratio of the length of the side adjacent to the angle to the hypotenuse:\[\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}\]This function aids in determining the horizontal component of a right triangle in relation to an angle.
Considerations of Cosine Function:
  • The cosine of an angle shows how much of the line segment is extending horizontally with respect to the hypotenuse.
  • This function is also bounded between -1 and 1 and is especially useful in calculating distances, projections, and understanding symmetry.
Tangent Function
The tangent function is the third primary trigonometric function and is abbreviated as \(\tan \theta\). It is derived from the sine and cosine functions and provides another important ratio:\[\tan \theta = \frac{\text{sine} \ \theta}{\text{cosine} \ \theta} = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite side}}{\text{adjacent side}}\]This definition means the tangent of an angle is the ratio of the opposite side to the adjacent side. It is especially useful in scenarios where we need to relate these two sides directly without reference to the hypotenuse.

Characteristics of the Tangent Function:

  • The tan function can take any real value, as it represents the ratio of two sides, and this ratio can increase without bound as \(\theta\) approaches certain values.
  • It's widely utilized in mathematics and various scientific fields where angle measures and side lengths need to be related directly.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the following steps to prove that \(\log _{b} x y=\log _{b} x+\log _{b} y\) a. Let \(x=b^{p}\) and \(y=b^{q} .\) Solve these expressions for \(p\) and \(q\) respectively. b. Use property E1 for exponents to express \(x y\) in terms of \(b, p\) and \(q\) c. Compute \(\log _{b} x y\) and simplify.

Large intersection point Use any means to approximate the intersection point(s) of the graphs of \(f(x)=e^{x}\) and \(g(x)=x^{123}\).

A cylindrical tank with a cross-sectional area of \(100 \mathrm{cm}^{2}\) is filled to a depth of \(100 \mathrm{cm}\) with water. At \(t=0,\) a drain in the bottom of the tank with an area of \(10 \mathrm{cm}^{2}\) is opened, allowing water to flow out of the tank. The depth of water in the tank at time \(t \geq 0\) is \(d(t)=(10-2.2 t)^{2}\). a. Check that \(d(0)=100,\) as specified. b. At what time is the tank empty? c. What is an appropriate domain for \(d ?\)

Let \(E\) be an even function and \(O\) be an odd function. Determine the symmetry, if any, of the following functions. $$E \circ E$$

Right-triangle relationships Use a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\tan ^{-1}\left(\frac{x}{\sqrt{9-x^{2}}}\right)\right)$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.