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Chapter 8: Problem 112

Write down the three Pythagorean Identities.

Short Answer

Expert verified
\( \sin^2(x) + \cos^2(x) = 1 \), \( \tan^2(x) + 1 = \sec^2(x) \), \( 1 + \cot^2(x) = \csc^2(x) \)

Step by step solution

01

Understand Pythagorean Identities

The Pythagorean Identities are derived from the Pythagorean Theorem in trigonometry. They are fundamental identities involving trigonometric functions.
02

First Pythagorean Identity

The first and most commonly known identity is derived from the unit circle definition: \[ \sin^2(x) + \cos^2(x) = 1 \]
03

Second Pythagorean Identity

Divide the first Pythagorean Identity by \( \cos^2(x) \): \[ \frac{\sin^2(x)}{\cos^2(x)} + 1 = \frac{1}{\cos^2(x)} \] This simplifies to: \[ \tan^2(x) + 1 = \sec^2(x) \]
04

Third Pythagorean Identity

Divide the first Pythagorean Identity by \( \sin^2(x) \): \[ 1 + \frac{\cos^2(x)}{\sin^2(x)} = \frac{1}{\sin^2(x)} \] This simplifies to: \[ 1 + \cot^2(x) = \csc^2(x) \]

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are equations that relate the trigonometric functions to one another. They are crucial tools in solving trigonometric problems because they allow us to simplify complex expressions and prove equivalencies.

The Pythagorean Identities, as the name suggests, are derived from the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. These identities are:

  • The first identity: \(\sin^2(x) + \cos^2(x) = 1\)
  • The second identity: \(\tan^2(x) + 1 = \sec^2(x)\)
  • The third identity: \(\tan^2(x) + 1 = \sec^2(x)\)
These identities are invaluable for simplifying expressions and solving equations in trigonometry.
Sine and Cosine Functions
The sine and cosine functions are among the most fundamental trigonometric functions. They originate from the unit circle, a circle with a radius of one centered at the origin of a coordinate plane. The coordinates of any point on this circle are given by (cos(x), sin(x)), where x is the angle formed with the positive x-axis.

  • Sine Function: The sine function, sin(x), represents the y-coordinate of the point on the unit circle at a given angle x.
  • Cosine Function: The cosine function, cos(x), represents the x-coordinate of the point on the unit circle at a given angle x.

The relationship \(\sin^2(x) + \cos^2(x) = 1\) is a result of the Pythagorean Theorem applied to the unit circle, as the hypotenuse (the radius) is always 1. Understanding these basic definitions helps in comprehending more complex trigonometric identities and functions.
Secant and Cosecant
Secant and cosecant are the reciprocals of the cosine and sine functions, respectively. They are less commonly used than sine and cosine, but they are still important in various trigonometric identities and equations.

  • Secant Function: The secant function, sec(x), is the reciprocal of the cosine function, defined as \(\sec(x) = \frac{1}{\cos(x)}\).
  • Cosecant Function: The cosecant function, csc(x), is the reciprocal of the sine function, defined as \(\csc(x) = \frac{1}{\sin(x)}\).

These functions come into play in the second and third Pythagorean Identities:

\(\tan^2(x) + 1 = \sec^2(x)\)
and
\(\tan^2(x) + 1 = \sec^2(x)\)
Understanding secant and cosecant allows you to manage a wider array of trigonometric problems and recognize various relationships between trigonometric functions.

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Most popular questions from this chapter

Solve each equation on the interval \(0 \leq \theta<2 \pi\). $$ \sin \theta-\cos \theta=-\sqrt{2} $$

If \(x=2 \tan \theta,\) express \(\cos (2 \theta)\) as a function of \(x .\)

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the area of the sector of a circle of radius 6 meters formed by an angle of \(45^{\circ}\). Give both the exact area and an approximation rounded to two decimal places.

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Simplify: \(\left(2 x^{2} y^{3}\right)^{4}\left(3 x^{5} y\right)^{2}\)

Challenge Problem Show that \(\tan ^{-1}\left(\frac{1}{v}\right)=\frac{\pi}{2}-\tan ^{-1} v\) if \(v>0\).
See all solutions

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